---
abstract: |
    We consider a reaction-diffusion system for two densities lying in adjacent domains of ${{\mathbb{R}}}^N$. We treat two configurations: either a cylinder and its complement, or two half-spaces. Diffusion and reaction heterogeneities for the two densities are considered, and an exchange occurs through the separating boundary.

    We study the long-time behavior of the solution, and, when it converges to a positive steady state, we prove the existence of an asymptotic speed of propagation in some specific directions. Moreover, we determine how such a speed qualitatively depends with respect to several parameters appearing in the model.

    In the case $N=2$, we compare such properties to those studied in @BRR1 [@BRR2; @BRR3; @BRR4] for a model with a line representing a road of fast diffusion at the boundary of a half-plane, which can be seen as a singular limit of the problem studied here.
author:
- 'Henri Berestycki[^1]'
- 'Luca Rossi[^2]'
- 'Andrea Tellini[^3]'
title: '**Coupled reaction-diffusion equations on adjacent domains**'
---

**Keywords:** Reaction-diffusion systems, Fisher-KPP equations, adjacent domains, diffusion heterogeneities, reaction heterogeneities, asymptotic speed of propagation, Robin eigenvalue problems, Bessel functions.

**2010 MSC:** 35K57, 35B40, 35K40, 35K45.

Introduction {#1section1}
============

In this paper we study the following system of two reaction-diffusion equations, one of which is set in the interior of a cylinder $\O:={{\mathbb{R}}}\times B_R(0)\subset{{\mathbb{R}}}^N$, $N\geq 2$, and the other one in its complement, i.e. $$\label{1eq:1.1}
\left\{
    \begin{array}{ll}
        u_t-D\D u=g(u) & \text{in $\O\times{{\mathbb{R}}}^+$} \\
        v_t-d\D v=f(v) & \text{in $({{\mathbb{R}}}^N\setminus\ov \O)\times{{\mathbb{R}}}^+$} \\
        D\, \p_n u = \nu v-\mu u & \text{on $\p\O\times{{\mathbb{R}}}^+$}\\
        -d\, \p_n v = \mu u-\nu v & \text{on $\p\O\times{{\mathbb{R}}}^+$,}
        \end{array}\right.$$ where we denote by $B_R(0)$ the ball of radius $R$ centered at the origin of ${{\mathbb{R}}}^{N-1}$. The constant $D>0$ represents the diffusion coefficient of $u(x,t)$, while $v(x,t)$ has a possibly different coefficient $d>0$, again constant.

The third and fourth equations of , in which $\p_n$ denotes the derivative in the outward normal direction to $\O$, describe an exchange given by a balanced flux through $\p\O$: a constant fraction $\mu>0$ of the density $u$ passes from $\O$ to its complement, while a constant fraction $\nu>0$ of $v$ goes from the complement to $\O$.

It is important to notice that the two densities are related only through this flux condition, and we do not impose any continuity ($u=v$) on the boundary. Instead, for every point $x_0\in\p\O$, $u(x_0,t)$ and $\p_n u(x_0,t)$ are defined by continuity on $\ov\O\times{{\mathbb{R}}}^+$, and, likewise, $v$ and $\p_n v$ are defined by continuity on $({{\mathbb{R}}}^N\setminus\O)\times{{\mathbb{R}}}^+$.

The reaction term $g$ will be assumed to be of Fisher-KPP type, i.e. a locally Lipschitz function, differentiable at $0$ and satisfying $$\label{eq:gKPP}
g(0)=g(1)=0, \qquad 0< g(s)\leq g'(0)s \,\text{ for  $s\in(0,1)$,} \qquad g(s)<0 \,\text{ for  $s>1$},$$ and, for some results, we will in addition require the following assumption, known as strong KPP property: $$\label{eq:gstrongKPP}
s\mapsto \frac{g(s)}{s} \quad \text{ is decreasing for $s>0$.}$$

Regarding the reaction term $f$, we will consider two possibilities: either Fisher-KPP again, i.e. we assume $f(s)$ to be a locally Lipschitz function satisfying $$\label{eq:fKPP}
f(0)=f(S)=0, \qquad 0< f(s)\leq f'(0)s \,\text{ for  $s\in(0,S)$,} \qquad f(s)<0 \,\text{ for  $s>S$}$$ for some $S>0$, and, in some cases, $$\label{eq:fstrongKPP}
s\mapsto \frac{f(s)}{s} \quad \text{ is decreasing for $s>0$,}$$ or we will take $f(s)=-\r s$ with $\r>0$, which amounts to put a mortality term in the complement of $\O$.

To sum up, system describes the spatial and temporal evolution of two densities in an environment which presents diffusion and reaction heterogeneities in two adjacent domains, with transmission (exchange) conditions at the interface of the two domains. Our main goal is to analyze the effect of such heterogeneities on the asymptotic behavior of the solutions of starting from a compactly supported initial datum $(u_0,v_0)$.

The motivation for studying such a system arises from the observation that some biological species or diseases diffuse or reproduce along specific directions faster than in the rest of the habitat. For example, the pine processionary caterpillar is believed to move faster on paths inside European forests (see @R), wolves in Western Canada preferentially use seismic lines (see @McK), and the early spread of HIV in the Democratic Republic of Congo was enhanced by transport networks like railways and rivers (see @Faria).

With the purpose of describing such situations, the series of works @BRR1 [@BRR2; @BRR3; @BRR4] has introduced the following *road-field* model with a line of fast diffusion at the boundary of a half-plane

$$\label{1eq:BRR2}
\left\{
\begin{array}{ll}
u_t-D\p^2_{x_1x_1} u=g(u)+\nu v-\mu u & \text{in ${{\mathbb{R}}}\times\{x_2=0\}\times{{\mathbb{R}}}^+$} \\
v_t-d\D v=f(v) & \text{in ${{\mathbb{R}}}\times\{x_2>0\}\times{{\mathbb{R}}}^+$} \\
-d\, \p_{x_2}v = \mu u-\nu v & \text{on ${{\mathbb{R}}}\times\{x_2=0\}\times{{\mathbb{R}}}^+$}, \end{array}\right.$$

and it has been proved (see @BRR2) that, under assumptions , , and , problem admits a unique positive, bounded steady state $(U,V(x_2))$, and that there exists a quantity ${c_{\mathrm{rf}}}>0$ such that

(i) \[th:1.1.i\] for all $c>{c_{\mathrm{rf}}}$, $\lim_{t\to\infty}\sup_{\substack{|x_1|\geq ct \\ x_2\geq 0}}(u,v)=(0,0)$,

(ii) \[th:1.1.ii\] for all $0<c<{c_{\mathrm{rf}}}$ and $a>0$, $\lim_{t\to\infty}\sup_{\substack{|x_1|\leq ct \\ 0\leq x_2\leq a}}|(u,v)-(U,V)|=0$.

These two properties amount to say that ${c_{\mathrm{rf}}}$ is the *asymptotic speed of propagation* of problem \[1eq:BRR2\] in the $x_1$ direction. Moreover, the qualitative properties of such a speed have been studied: precisely, it has been showed that, if we set $c_f:=2\sqrt{df'(0)}$, $$\label{1eq:5.1bis}
{c_{\mathrm{rf}}}=c_f \quad \text{if $\frac{D}{d}\leq 2-\frac{g'(0)}{f'(0)}$,} \qquad \text{while} \qquad {c_{\mathrm{rf}}}>c_f \quad \text{otherwise,}$$ that, in the latter case, the function $D\mapsto {c_{\mathrm{rf}}}(D)$ is increasing and, in addition, $$\label{1eq:5.1ter}
\lim_{D\to+\infty}\frac{{c_{\mathrm{rf}}}(D)}{\sqrt{D}}\in(0,+\infty)$$ (in particular $\lim_{D\to+\infty}{c_{\mathrm{rf}}}(D)=+\infty$).

We recall that the quantity $c_f$ is the asymptotic speed of propagation (in any direction) for the homogeneous Fisher-KPP equation $v_t-d\D v=f(v)$ in ${{\mathbb{R}}}^N$, $N\geq 1$ (see @F [@KPP; @AW]). As a consequence, establishes for which values of the parameters the heterogeneity introduced in by the presence of the road enhances the propagation speed, while shows that such an enhancement becomes arbitrarily large when a sufficiently large value of the diffusion $D$ is considered on the line.

With respect to , our configuration in with $N=2$ represents the case in which the road of fast diffusion is replaced by a *strip*. Therefore, the goal of this work is to study whether the properties of the road-field model studied in @BRR2 still hold true with a thick region, and what is the behavior for higher spatial dimensions. In particular, we prove the existence of an asymptotic speed of propagation, and we study when an enhancement with respect to the homogeneous case takes place, as well as the behavior of the asymptotic speed of propagation as the diffusion $D$ and the radius of the cylinder vary. Our first result is the following.

\[1th:3.3\] Assume , , , and let $(u_0,v_0)$ be non-negative, bounded and continuous. Then, there exists a steady state $(U,V)$ of such that the solution $(u,v)$ of with $(u_0,v_0)$ as an initial datum satisfies $\lim_{t\to+\infty}(u,v)=(U,V)$.

Moreover, for $2\leq N\leq 5$, if $(u_0,v_0)$ has compact support, there exists $c^*>0$ satisfying:

(i) \[1th:3.3.i\] for all $c>c^*$, $\lim_{t\to\infty}\sup_{|x_1|\geq ct}(u,v)=(0,0)$, uniformly in $(x_2,\ldots,x_N)\in{{\mathbb{R}}}^{N-1}$;

(ii) \[1th:3.3.ii\] for all $0<c<c^*$, $\lim_{t\to\infty}\sup_{|x_1|\leq ct}|(u,v)-(U,V)|=0$, locally uniformly in $(x_2,\ldots,x_N)\in{{\mathbb{R}}}^{N-1}$.

In addition, $c^*\geq c_f$, and the following qualitative properties hold true:

(iii) \[1th:3.3.iii\] there exists $C=C(D,d,N,R,\mu,\nu)\geq 0$ such that $$\label{1eq:enhancement}
        c^*>c_f \quad \text{if and only if} \quad \frac{D}{d}>2-\frac{g'(0)}{f'(0)}+C.$$ For $N=2,3$ $C=0$, otherwise $C>0$;

(iv) \[1th:3.3.iv\] for $D\to+\infty$, $c^*(D)/\sqrt{D}$ converges to a positive real number;

(v) \[1th:3.3.v\] When holds true for $D\sim 0$, then $$\label{1eq:4.5}
        \lim_{D\to 0} c^*(D)=:c_0=g'(0)\sqrt{\frac{d}{g'(0)-f'(0)}}\,\begin{cases}
        =c_f & \text{if } g'(0)=2f'(0) \\
        >c_f & \text{if } g'(0)>2f'(0).
        \end{cases}$$ Otherwise, $\lim_{D\to 0} c^*(D)=c_f$;

(vi) \[1th:3.3.vi\] the function $R\mapsto c^*(R)$ is strictly increasing whenever $c^*>c_f$. Moreover, it satisfies: $$\label{0eq:1.10}
            \lim_{R\da 0} c^*(R)=c_f, \qquad \lim_{R\to+\infty}c^*(R)=c^*_{\infty}:=
            \begin{cases}
            c_f & \text{ if $\frac{D}{d}\leq 2-\frac{g'(0)}{f'(0)}$} \\
            c_g & \text{ if $\frac{d}{D}\leq 2-\frac{f'(0)}{g'(0)}$} \\
            c_{a} & \text{ otherwise,}
            \end{cases}$$ where, in analogy with the notation used above, $c_g:=2\sqrt{Dg'(0)}$ denotes the Fisher-KPP speed corresponding to the diffusion and reaction inside $\O$, while $$\label{0eq:1.11}
            c_{a}:=\frac{|Df'(0)-dg'(0)|}{\sqrt{(D-d)(f'(0)-g'(0))}}.$$

As we see from part of the previous result, the spatial dimension has a direct effect on the threshold for the enhancement of the asymptotic speed of propagation, since such a threshold is strictly larger than the road-field one when $N=4,5$. The limitation $N\leq 5$ in Theorem \[1th:3.3\] is due to a technical point in the construction of the sub- and supersolutions that allow us to characterize $c^*$, and it is related to the singular behavior of the modified Bessel function of second kind, as it will be apparent in Section . However, we will be able to determine when the asymptotic speed of propagation coincides with $c_f$ in more general situations (see Remark \[1re:4.2\]), and we will prove (see Proposition \[1pr:4.2bis\]) that it is the case, for example, for sufficiently large $N$. This means that the cylinder looses its effect for large dimensions, which can be explained since the relative volume of $\O$ with respect to any cylinder with radius $R'>R$ goes to $0$ as the dimension goes to $+\infty$.

In contrast with , in Theorem \[1th:3.3\] we do not give any monotonicity result of the asymptotic speed of propagation with respect to $D$. Indeed, we will show that in some cases it is not monotonic (see Remark \[1re:4.2\]), which, up to our knowledge, is a phenomenon established in this work for the first time.

Passing to comment part , it establishes that, also in the limit $R\da 0$, the cylinder does not have any effect on the propagation, thus there is no direct relation of , considered with $N=2$, with the road-field system . Nonetheless, we will prove that, up to performing a singular rescaling in the exchange coefficient $\mu$, we recover the road-field speed of propagation ${c_{\mathrm{rf}}}$ as $R\da 0$. The precise result is the following.

\[1th:5.1\] Problem $$\label{1eq:5.2}
    \left\{
    \begin{array}{ll}
    u_t-D\D u=g(u) & \text{in ${{\mathbb{R}}}\times(-R,R)\times{{\mathbb{R}}}^+$} \\
    v_t-d\D v=f(v) & \text{in ${{\mathbb{R}}}\times\{|y|>R\}\times{{\mathbb{R}}}^+$} \\
    \pm D\, \p_{x_2}u = \nu v-\tilde\mu(R) u & \text{on ${{\mathbb{R}}}\times\{y=\pm R\}\times{{\mathbb{R}}}^+$}\\
\mp d\, \p_{x_2}v= \tilde\mu(R) u-\nu v & \text{on ${{\mathbb{R}}}\times\{y=\pm R\}\times{{\mathbb{R}}}^+$}, 
    \end{array}\right.$$ where $\tilde{\mu}(R)$ is a positive function satisfying $$\label{1eq:5.2bis}
    \lim_{R\da0}\frac{\tilde{\mu}(R)}{R}=\mu,$$ $\mu$ being the parameter appearing in problem , admits an asymptotic speed of propagation in the $x_1$ direction $\tilde c^*(R)$ which converges, as $R\to 0$, to ${c_{\mathrm{rf}}}$, the asymptotic speed of propagation of problem in the $x_1$ direction.

In addition, we will study the behavior of $\tilde c^*(R)$ when other rescalings, different from , are performed (see Proposition \[1pr:5.2\]).

Regarding the limit as $R\to+\infty$ we will show that it coincides with the asymptotic speed of propagation in the $x_1$ direction of the following problem posed in adjacent half-spaces: $$\label{0eq:1.1}
\left\{
\begin{array}{ll}
u_t-D\D u=g(u) & \text{in $\{x_N<0\}\times{{\mathbb{R}}}^+$} \\
v_t-d\D v=f(v) & \text{in $\{x_N>0\}\times{{\mathbb{R}}}^+$} \\
D\, \p_{x_N} u = \nu v-\mu u & \text{on $\{x_N=0\}\times{{\mathbb{R}}}^+$}\\
-d\, \p_{x_N} v = \mu u-\nu v & \text{on $\{x_N=0\}\times{{\mathbb{R}}}^+$}
\end{array}\right.$$ (actually, due to the symmetries of such a problem, the speed of propagation is the same in any direction satisfying $x_N=0$). This establishes a continuous dependence of the asymptotic speed of propagation or problem with respect to the domain. Indeed, one can think one point of the boundary $\p\O$ to be fixed; thus, as $R\to+\infty$ and the curvature of the cylinder converges to $0$, one part of the boundary approaches the hyperplane which separates the half-spaces, while the rest of the boundary disappears at $+\infty$. In fact, to prove this result, we will first construct the asymptotic speed of propagation for problem , then the one for problem and, comparing both constructions, we will show the latter converges to the former.

In the final part of the paper (Section \[section4\]), we analyze the case in which the reaction $f$ describes a (linear) mortality term in the complement of $\O$, i.e. we consider the following problem $$\label{2eq:4.0} 
\left\{
\begin{array}{ll}
u_t-D\D u=g(u) & \text{in $\O\times{{\mathbb{R}}}^+$} \\
v_t-d\D v=-\r v & \text{in $({{\mathbb{R}}}^N\setminus\overline{\O})\times{{\mathbb{R}}}^+$} \\
D\, \p_n u = \nu v-\mu u & \text{on $\p B_R(0)\times{{\mathbb{R}}}^+$}\\
-d\, \p_n v = \mu u-\nu v & \text{on $\p B_R(0)\times{{\mathbb{R}}}^+$.}
\end{array}\right.$$ A motivation for treating also this kind of reaction terms is given by the propagation of Scentless Chamomile in the region of Saskatchewan in Western Canada, which, being transported by agricultural vehicles, spreads faster along roads, but faces the competition of hostile weeds in the surrounding fields (see @dCL). Once again, we want to compare the situation in which $u$ occupies a thick domain with the analogous road-field version in a half-plane, which has been considered, with $g\equiv 0$, in @BRR4. First, we state a summary of our results - we send the reader to Section \[section4\] for a complete perspective -, and then we compare them with the road-field case.

\[2th:section4.1\] Assume and . Then, the solution of starting from a non-negative, bounded, continuous $(u_0,v_0)$ converges to a positive steady-state of as $t\to+\infty$ if and only if $$\label{2eq:condnecsuff}
    \frac{g'(0)}{D}>\b_0^2,$$ where $\b_0^2=\b_0^2(D,d,N,R,\mu,\nu,\r)$ is the principal eigenvalue of the Robin eigenvalue problem . Otherwise, the solution converges to $(0,0)$, uniformly in each domain.

Moreover, in the former case, assuming in addition that $N\leq 5$ if $D<d$, and $(u_0,v_0)$ to have compact support, there exists and asymptotic speed of propagation, denoted by $c^*_m$.

In @BRR4 [Theorems (2.5) and (2.6)] a necessary and sufficient condition for invasion was established also for the road-field analogue. As it is natural to expect, the condition there does not depend neither on $R$, which is not present, nor on $N$, which is fixed to be equal to 2. Nonetheless, it does not depend on $D$ either, as it is the case in instead.

In this work, we establish when holds true as these quantities, taken one by one, vary. When it is the case, we focus on the qualitative behavior of the asymptotic speed $c^*_m$. The results are summarized in the following theorem.

\[2th:section4.2\]

(i) \[2th:section4.2.i\] Condition does not hold true for sufficiently large $N$. As a consequence, under the assumptions of Theorem \[2th:section4.1\], the solution of starting with a non-negative, bounded initial datum $(u_0,v_0)$ satisfies $\lim_{t\to+\infty}(u,v)=(0,0)$, uniformly in each domain.

(ii) \[2th:section4.2.ii\] There exists $R_0>0$ such that holds true if and only if $R>R_0$. Moreover, the function $R\mapsto c^*_m(R)$ is increasing and satisfies $$\label{1eq:1.13}
            \lim_{R\da R_0} c^*_m(R)=0, \qquad \lim_{R\to+\infty}c^*_m(R)=c^*_{m,\infty}:=
            \begin{cases}
            c_g & \text{ if $\frac{d}{D}\leq 2+\frac{\r}{g'(0)}$} \\
            c_{m,a} & \text{ otherwise,}
            \end{cases}$$ where $$\label{1eq:1.14}
            c_{m,a}:=\frac{D\r+dg'(0)}{\sqrt{(d-D)(\r+g'(0))}}.$$

(iii) \[2th:section4.2.iii\] If $$\label{2eq:6.2}
            \frac{\mu(N-1)}{R\, g'(0)}\leq 1+\frac{\nu}{\sqrt{d\r}}\frac{K_{\tau}\left(\sqrt{\frac{\rho}{d}}R\right)}{K_{\tau+1}\left(\sqrt{\frac{\rho}{d}}R\right)},$$ where $K_\tau$ is the modified Bessel function of second kind, then holds true for every $D>0$. Moreover,

    -   \[2th:6.2iv\] if strict inequality holds true in , then $\lim_{D\to+\infty}c^*_m(D)/\sqrt{D}\in(0,+\infty)$;

    -   \[2th:6.2v\] if equality holds true in , then $\limsup_{D\to+\infty}c^*_m(D)<+\infty$.

    Otherwise, if does not hold true, there exists $D_0\in(0,\infty)$ such that holds true if and only if $D<D_0$, and $\lim_{D\ua D_0}c^*_m(D)=0$.

Part of the previous result is a counterpart for this problem of the fact that, for large $N$, the cylinder does not enhance the propagation speed of , since its effect becomes negligible with respect to the complement. The difference here is that, since the exterior domain is hostile, the good environment provided by $\O$ cannot prevent the densities to go extinct.

Along the same line, part establishes that, all the other parameters being fixed, the densities can survive only if the good environment has a sufficiently large section. In addition, in analogy with problem , the larger the section of $\O$, the larger the asymptotic speed of propagation, and the limit as $R\to+\infty$ will be characterized as the asymptotic speed of spreading of the problem corresponding to in two adjacent half-spaces.

To conclude the analysis of our results, we observe from part that, on the one hand, when does not hold true, we obtain, for $D\sim D_0$, another example in which the asymptotic speed of propagation is not monotone in $D$. On the other hand, we see that the role of $D$ is more subtle, even in the case in which $c^*_m(D)$ is defined for all $D$, since the asymptotic speed of propagation can be bounded as $D\to+\infty$, which was not the case in and in all the road-field problems treated in the literature before.

Once presented the results of this paper, we conclude the introduction by relating them with those of other previous works.

**Related works and further comments.** The notation $c_a$ used in stands for *anomalous speed*. Such a name has been introduced in @WLL in the context of some cooperative systems of equations set in the same domain (see also @H for a system of two coupled Fisher-KPP equations again in the same domain) and reflects the fact that such speed of propagation is greater than both of the Fisher-KPP speeds at which each density would invade its domain if it was isolated - incidentally, this will be also apparent from the construction of $c^*_\infty$ in Section \[0section:2.2\] (see Proposition \[0pr:1.6\]). Analogously, we have used the notation $c^*_{m,a}$ in - for the anomalous speed of propagation arising in the limit $R\to+\infty$ for problem .

It turns also out that $c^*_\infty$ coincides with the asymptotic speed of propagation of another system, recently studied in @BCM, describing the evolution of two densities that represent two parts of a population with different phenotypes. Such subpopulations are assumed to compete between each other, and to diffuse and grow each with its own rate. A difference with is that in @BCM the two densities share the same environment and can mutate from one type to the other. The rates of mutation play the role of the exchange coefficients that in our system allow each of the densities to pass to the adjacent domain.

Passing to the work @BN, as an application of the main result, a unique reaction-diffusion equation has been considered in the plane, with Fisher-KPP reaction $f$ and discontinuous diffusion coefficients: $d$ in the upper half-plane and $D$ in the lower one. At least formally, this corresponds to consider for $N=2$ and let $\mu=\nu\to+\infty$, even if the rigorous treatment of such a limit and the relation between the two problem is a very interesting open question which deserves further work. In @BN, the authors have determined the asymptotic speed of propagation in every direction and, in particular, they have proved that, in the $x_1$ direction, it coincides with the maximum of the two Fisher-KPP speeds $c_f$ and $c_g$, as it happens for the asymptotic speed of , $c^*_\infty$, if $f'(0)=g'(0)$, as it can be seen from the diagram in Figure \[0fig:1.1\].

Regarding the result of Theorem \[1th:5.1\], in @LW, the authors have considered a similar situation with a strip whose thickness goes to $0$ and, by performing a singular rescaling to the diffusivity $D$, they have obtained a problem with effective boundary conditions on a line, which exhibits an enhanced speed of propagation with respect to the Fisher-KPP one. However, the problem in @LW is essentially different from the one here, since the model there has only one density and discontinuous diffusion coefficients. Moreover, the speed of propagation of the limiting problem is not related to the speed ${c_{\mathrm{rf}}}$ of , as it is the case in Theorem \[1th:5.1\], where the singular rescaling is performed on $\mu$ instead.

As already commented above, the limitation in the spatial dimension that arises in Theorems \[1th:3.3\] and \[2th:section4.1\] is due to the singularity at $0$ of the modified Bessel functions of second kind $K_\tau$, that we use because the domain has axial symmetry with an unbounded component. This was not the case in @RTV, where a Fisher-KPP reaction inside a cylinder in ${{\mathbb{R}}}^N$, with a different diffusion on the boundary has been studied. In such work, since the section of the domain was bounded, it was not necessary to use the functions $K\tau$, and a precise asymptotics speed of propatagion has been determined for all $N\geq 2$. We believe that the limitation here is only technical and that it can be overcome by constructing other supersolutions which allow to continue the curves in Sections and \[2section4\], in the spirit of the proof of Proposition \[0pr:1.5\], or, alternatively, by using completely different approaches.

The first mortality condition in the context of road-field models has been introduced in @T16, where a system like , with $g(u)\equiv 0$ and the field consisting in the strip ${{\mathbb{R}}}\times(0,R)$ has been studied, obtaining a necessary and sufficient condition for invasion to occur. However, differently from and @BRR4, the mortality condition in @T16 was not modeled through the reaction term, but through the boundary condition $v=0$ at $x_2=R$.

Finally, we mention other works that treat systems of reaction-diffusion equations posed in a domain and on its boundary, in the vein of road-field models, but with a bounded domain: @MCV is focused on the phenomenon of the formation of Turing patterns in the interior and/or on the boundary of the domain, while @FLT considers a nonlinear coupling between the domain and its boundary, and proves the exponential convergence of the solution to the equilibrium.

**Structure of the paper.** The paper is distributed as follows: in Section \[0section1\] we study , i.e., the case of two adjacent half-spaces with Fisher-KPP nonlinearities in both of them; in Section \[section3\] we pass to our main problem , i.e., we consider the same type of nonlinearities in a cylinder and its complement. Later, in Section \[section4\], we treat the mortality case for $v$ in both configurations of the domain. Finally, in Appendix \[sectionapp1\], we recall some facts on Bessel functions that arise in the study of the cylindrical case, and we prove some results related to them which are of independent interest.

Two adjacent half-spaces with Fisher-KPP nonlinearities {#0section1}
=======================================================

For the sake of clearness in the exposition, we start by studying system with $g$ and $f$ both of Fisher-KPP, i.e. satisfying and , respectively. Indeed, for such a problem we will not need to introduce the Bessel functions which, on the contrary will appear in the treatment of the cylindrical domain. Moreover, the construction itself of the asymptotic speed of propagation for problem will be necessary later to prove some of the qualitative properties of the one of .

In the first part of the section we study the long-time behavior of the solutions of such a system, while in the second part we determine the asymptotic speed of propagation, and we prove some of its properties.

Long-time behavior for system 
------------------------------

In order to determine the asymptotic of the solutions as $t\to+\infty$, we will use some comparison results, which can be shown to hold true with the same techniques of @BRR1 [Proposition 3.2], to prove the existence, uniqueness and some symmetries of positive steady states, i.e. solutions of which do not depend on $t$. We start with the following result.

\[0pr:1.1\] Assume and . Then, for every continuous, bounded and non-negative $(u_0,v_0)$, there exist two positive bounded stationary solutions $(U_i,V_i)$ of , $i\in\{1,2\}$, which depend only on $x_N$ and such that the solution of starting from $(u_0,v_0)$ satisfies $$\label{0eq:1.2}
    (U_1,V_1)\leq\liminf_{t\to+\infty}(u,v)\leq\limsup_{t\to+\infty}(u,v)\leq(U_2,V_2)$$ locally uniformly in each domain.

If we set $$\label{0eq:1.2bis}
        \ov{u}:=\max\left\{1, \|u_0\|_{\infty},\frac{\nu}{\mu}S, \frac{\nu}{\mu}\|v_0\|_{\infty}\right\}, \qquad \ov{v}:=\frac{\mu}{\nu}\ov{u},$$ $(\ov u,\ov v)$ is a (positive) stationary supersolution of . Thanks to the parabolic strong comparison principle, the solution of starting from $(\ov u,\ov v)$ is decreasing in time, and - being positive, thus bounded - it converges to a stationary state $(U_2,V_2)$. Moreover, since $(\ov u,\ov v)$ lies above $(u_0,v_0)$, the solution of with $(u_0,v_0)$ as an initial datum maintains the same order for all $t>0$. Taking the $\limsup$ as $t\to+\infty$, the last inequality in is therefore proved. Fix now any $i\in\{1,\ldots,N-1\}$; as the initial datum $(\ov u,\ov v)$ does not depend on $x_i$, the same property holds true for the unique solution of , thanks to the invariance of the problem by translations with respect to $x_i$. Ultimately, the limit $(U_2,V_2)$ inherits the same symmetries.

In order to find the other steady state, we perform quite a classical construction of a stationary subsolution with compact support. To this end, consider the eigenvalue problem $$\label{0eq:1.3}
        \left\{
        \begin{array}{ll}
        -\D \phi=\l \phi & \text{ in $B_L:=B((0,\ldots,0,-(L+1)),L)$} \\
        \phi=0 & \text{ on $\p B_L$,}
        \end{array}\right.$$ where $B((0,\ldots,0,-(L+1)),L)$ denotes the ball of radius $L>0$ and center $(0,\ldots,0,-(L+1))\in\{x_N<0\}$. It is well known that admits a smallest eigenvalue $\l=\l_1(L)$, which is positive and to which a positive eigenfunction $\phi_1$, that will be normalized so that $\max_{B_L}\phi_1=1$, is associated. Moreover $\l_1(L)\to 0$ as $L\to+\infty$, thus, if we take a sufficiently large $L$ so that $\l_1(L)<\frac{g'(0)}{D}$, there exists $\e_0>0$ such that, by setting $$        \un u:=\left\{
        \begin{array}{ll}
        \phi_1(x) & \text{ for $x\in B_L$} \\
        0 & \text{ for $x\in\{x_N<0\}\setminus B_L$,}
        \end{array}\right.$$ $\e(\un u,0)$ is a strict non-negative stationary subsolution of for all $\e\in(0,\e_0)$. Reasoning as above, we have that the solution of starting from $\e(\un u,0)$ converges, now strictly increasing, towards a positive steady state $(U_1,V_1)$. Moreover, up to reducing $\e_0$ if necessary, we can assume that $\e(\un u,0)$ lies strictly below the solution $(u,v)$ evaluated at $t=1$. The comparison principle thus ensures that the order is preserved, and, by taking the $\liminf$ as $t\to+\infty$, we obtain the first inequality in .

To conclude, we need to prove that $(U_1,V_1)$ is $x_i$-independent for every $i\in\{1,\ldots,N-1\}$. Since the domain and the system are invariant by rotations around the axis $\{x_1=\ldots=x_{N-1}=0\}$, it is sufficient to prove the invariance with respect to $x_1$. As $\e(\un u,0)$ has compact support and the solution starting from it is increasing in time, we have that it lies strictly below $(U_1,V_1)$. Thus, that there exists $h_0>0$, $h_0\sim 0$, such that $\e(\un{u}(x_1-h,x_2,\ldots,x_N),0)$, which is still a subsolution of by the invariance of the problem by translations in $x_1$, lies below $(U_1(x_1,\ldots,x_N),V_1(x_1,\ldots,x_N))$ for all $h\in(-h_0,h_0)$. From the uniqueness of the Cauchy problem associated to , we obtain that the solution of the system, with the translated subsolution as initial datum, converges to the corresponding translation of $(U_1,V_1)$. Thus, by comparison, $(U_1,V_1)$ is smaller than small translations in the $x_1$ direction of itself, which proves that it does not depend on $x_1$.

Thanks to the previous result, we are lead to consider bounded steady states which depend only on $x_N$, for which we obtain the following proposition.

\[0pr:1.2\] Assume and . Then, there exists a unique positive, bounded steady state of problem that does not depend on $x_1,\ldots,x_{N-1}$. Such a steady state will be denoted by $(U,V)$.

The existence part has been proved in Proposition \[0pr:1.1\], thus we focus on uniqueness. Using the symmetries of the steady states that we are considering, reduces to the following system of ordinary differential equations $$\label{0eq:1.5}
    \left\{
    \begin{array}{ll}
    -D U''(y)=g(U(y)) & \text{in $\{y<0\}$} \\
    -d V''(y)=f(V(y)) & \text{in $\{y>0\}$} \\
    D\, U'(0) = \nu V(0)-\mu U(0) & \\
    -d\, V'(0) = \mu U(0)-\nu V(0), & 
    \end{array}\right.$$ for which we will show that there exists at most one bounded positive solution. Thanks to the assumptions on $f$ and $g$, the origin is a center in the phase planes associated to each of the differential equations, thus any positive bounded solution must satisfy $$    U(-\infty)=1, \quad U'(-\infty)=0, \quad V(+\infty)=S, \quad V'(+\infty)=0.$$ We will now prove that one of the following mutually exclusive possibility occurs in the whole domain of definition of the functions:

-   $U$ and $V$ are both decreasing, $0<U<1$ and $V>S$,

-   $U$ and $V$ are both constant, $U=1$ and $V=S$,

-   $U$ and $V$ are both increasing, $U>1$ and $0<V<S$.

Indeed, we distinguish three cases according to the value of $U(0)$: $0<U(0)<1$, $U(0)=1$, or $U(0)>1$. In the former one, if by contradiction there exists $y_0<0$ for which $U(y_0)\geq 1$, then, since $U(-\infty)=1$, $U$ will have a maximum value equal to or greater than $1$. Such a maximum cannot be equal to $1$, because, at any maximum point $y_M$, $U'(y_M)=0$, and the uniqueness of the Cauchy problem associated to the differential equation for $U$ will imply $U\equiv 1$, against our assumption on $U(0)$. If the maximum was greater than $1$, then, at any maximum point, $$0\geq D U''(y_M)=-g(U(y_M))>0,$$ again a contradiction. Once we know that $0<U<1$ for every $y<0$, the sign of $g$ entails that $g$ is concave and, since $U'(-\infty)=0$, $U$ is decreasing.

The third and fourth equations of give that $dV'(0)=DU'(0)<0$, thus, for $V$ not to vanish somewhere, the sign of $f$ forces $V(0)>S$ and, by reasoning as we have done for $U$, it follows that $V>1$ everywhere, and that it is decreasing.

Analogously, it is possible to show that if $U(0)=1$ (respectively $U(0)>1$) the second (respectively the third) possibility occurs.

To conclude the proof, we show that in any of the three cases, system admits at most one solution. Let us start with the first case: by multiplying the differential equations of by $U'$ and $V'$ respectively and then integrating, we obtain that the quantities $$\label{0eq:1.6bis}
        \frac{1}{2}U'^2(y)+\frac{1}{D}\int_{U(0)}^{U(y)}g(s)\,ds \quad \text{ and } \quad \frac{1}{2}V'^2(y)+\frac{1}{d}\int_{V(0)}^{V(y)}f(s)\,ds$$ are constant for every $y$ and, evaluating them at the extrema of their respective domains, we get the following identities $$\label{0eq:1.6ter}
    \int_{U(0)}^{1}g(s)\,ds=\frac{D}{2}U'(0)^2, \qquad  \int_{V(0)}^{S}f(s)\,ds=\frac{d}{2}V'(0)^2.$$ The left-hand side of the first equation is decreasing with respect to $U(0)$, thus so is $U'(0)^2$ and, as $$\label{0eq:1.7}
    \left\{
    \begin{array}{l}
    d\,V'(0)=D\, U'(0)  \\
    \nu V(0)=\mu U(0)+dV'(0)=\mu U(0)+DU'(0),
    \end{array}\right.$$ $V'(0)^2$ is decreasing. Since $V'(0)$ is negative, it is an increasing function of $U(0)$, and the second relation of gives that $V(0)$ also increases with $U(0)$. Finally, we observe that $\int_{V(0)}^Sf(s)\,ds=\int_S^{V(0)}(-f(s))\,ds$, thus these integrals are increasing with respect to $U(0)$. These considerations entail that the second relation in has at most one solution as $U(0)$ varies in $(0,1)$.

In the second case it is easy to see that, since $1$ and $S$ are the unique positive zeros of $g$ and $f$ respectively, the unique solution of is $(U,V)\equiv (1,S)$.

Finally, the third case can be treated as the first one, by interchanging the roles of $U$ and $V$ and showing that the first relation in has at most one solution as $V(0)$ varies in $(0,S)$.

By combining the two previous results, we get the following long-time behavior for system .

\[0th:1.3\] Assume and . Then, for every continuous, bounded and non-negative $(u_0,v_0)$, the solution of starting from $(u_0,v_0)$ satisfies $$\lim_{t\to+\infty}(u,v)=(U,V)$$ locally uniformly in each domain, where $(U,V)$ is the steady state given by Proposition \[0pr:1.2\].

Asymptotic speed of propagation for  {#0section:2.2}
------------------------------------

In this section we construct super- and subsolutions to moving with certain speeds that will provide us with upper and lower bounds, respectively, for the asymptotic speed of propagation for problem in the $x_i$ direction, $i\in\{1,\ldots,N-1\}$. Thanks to the symmetry of the problem under rotations around the axis $\{x_1=\ldots=x_{N-1}=0\}$, it will suffice to consider the positive $x_1$ direction; accordingly, we will detail all the constructions in such a case.

In order to obtain supersolutions, we consider the linearization of around $(0,0)$, which reads $$\label{0eq:1.8}
    \left\{
    \begin{array}{ll}
        u_t-D\D u=g'(0)u & \text{in $\{x_N<0\}\times{{\mathbb{R}}}^+$} \\
        v_t-d\D v=f'(0)v & \text{in $\{x_N>0\}\times{{\mathbb{R}}}^+$} \\
        D\, \p_{n} u = \nu v-\mu u & \text{on $\{x_N=0\}\times{{\mathbb{R}}}^+$}\\
        -d\, \p_{n} v = \mu u-\nu v & \text{on $\{x_N=0\}\times{{\mathbb{R}}}^+$,}
    \end{array}\right.$$ and we look for the values of $c$ for which such a system admits (super)solutions of the following type $$\label{0eq:1.9}
    (\ov u,\ov v)=e^{\mp\a(x_1\mp ct)}\left(1,\frac{\mu}{\nu}\right),$$ where $\a$ is a positive constant to be determined. Indeed, thanks to the Fisher-KPP hypothesis and , (super)solutions of provide us with supersolutions of the nonlinear problem . The result is the following.

\[0pr:1.2bis\] Let $c^*_\infty$ the quantity defined in . Then, problem admits supersolutions of the form for every $c\geq c^*_\infty$.

Observe that the functions of the form satisfy the third and the fourth equations in . Thus, by plugging into the system, we have that such a guess is a supersolution if and only if $$\left\{
        \begin{array}{l}
            D\a^2-c\a +g'(0)\leq 0 \\
            d\a^2-c\a+f'(0)\leq 0.
        \end{array}\right.$$ This shows that we have to look, according to the different values of $D, d, g'(0), f'(0)$, for the values of $c>0$ for which the parabolas $$\Si_D(c):=\{D\a^2-c\a +g'(0)=0\}, \qquad \Si_d(c):=\{d\a^2-c\a +f'(0)=0\},$$ have real roots and the regions on the positive $\a$ axis lying between such roots intersect. The roots are given, respectively, by $$\label{0eq:1.12bis}
 r_D^{\pm}(c):=\frac{c\pm\sqrt{c^2-c_g^2}}{2D}, \qquad r_d^{\pm}(c):=\frac{c\pm\sqrt{c^2-c_f^2}}{2d},$$ thus we observe that, for them to be real, necessarily $c\geq\max\{c_f,c_g\}$. In addition, the graphs of the parabolas decrease in the region $\{\a>0\}$ as $c$ increases. After these preliminary observations, we distinguish several cases and subcases.

*Case 1 ($c_f<c_g$).* In this case, for $c=c_f$, the parabola $\Si_d(c)$ intersects the axis $\{\a>0\}$ but $\Si_D(c)$ does not, and the same configuration persists until $c=c_g$, when the parabola $\Si_D$ touches the axis for the first time. We distinguish three cases according to the relative position of $\Si_d(c)$ and $\Si_D(c)$ for $c=c_g$:

(i) if $r_d^-(c_g)\leq r_D^{\pm}(c_g)=\frac{c_g}{2D}\leq r_d^+(c_g)$, which can be equivalently written as $$c_g\left|\frac{1}{2D}-\frac{1}{2d}\right|\leq\frac{\sqrt{c^2_g-c_f^2}}{2d},$$ and it is easy to see that such a relation holds true if and only if the second condition in is satisfied, then the monotonicity of the parabolas with respect to $c$ entails that the regions under consideration intersect for every $c\geq c_g$;

(ii) if $r_D^{\pm}(c_g)> r_d^+(c_g)$, there is are no intersections on the axis for $c=c_g$ and, in order to get them, $c$ has to be increased up to the value such that $r_D^{-}(c)= r_d^+(c)$, which equals $c_a$;

(iii) similarly, if $r_D^{\pm}(c_g)< r_d^+(c_g)$, we have intersections for every $c$ starting from the value satisfying $r_D^{+}(c)= r_d^-(c)$, which is again $c_a$.

*Case 2 ($c_f=c_g$).* In this case, the parabolas $\Si_d(c)$ and $\Si_D(c)$ touch the axis at the same time for $c=c_f=c_g$ and we have to distinguish three cases according to the relative position of such double roots:

(i) if $r_d^\pm(c_g)=r_D^{\pm}(c_g)$, which is equivalent to $d=D$ and $f'(0)=g'(0)$, we have that $\Si_d(c)$ and $\Si_D(c)$ coincide for every $c$, thus intersection occurs for every $c\geq c_f=c_g$;

(ii) if $r_d^\pm(c_g)>r_D^{\pm}(c_g)$, which holds true if and only if $d<D$ (and, as a consequence $f'(0)>g'(0)$), we have intersection starting from the value of $c$ for which $r_d^-(c)=r_D^+(c)$, which coincides again with $c_a$;

(iii) if $r_d^\pm(c_g)<r_D^{\pm}(c_g)$, then the smallest value of $c$ for having the desired intersections is the one for which $r_d^+(c)=r_D^-(c)$, again $c_a$.

*Case 3 ($c_f>c_g$).* This case can be handled by reasoning as in case 1, by interchanging the roles of the parameters in the upper and the lower half-planes, which explains also the emergence of the condition in the first line of , which is obtained from the condition in the second line by replacing $d$ with $D$, $f'(0)$ with $g'(0)$, and vice versa.

We now pass to the construction of compactly supported stationary subsolutions for the following auxiliary problem, which has an additional transport term $cu_{x_1}$, for speeds $c<c^*_\infty$, $c\sim c^*_\infty$: $$\label{0eq:1.13}
    \left\{
    \begin{array}{ll}
        u_t-D\D u+ cu_{x_1}=g(u) & \text{in $\{x_N<0\}\times{{\mathbb{R}}}^+$} \\
        v_t-d\D v+ cv_{x_1}=f(v) & \text{in $\{x_N>0\}\times{{\mathbb{R}}}^+$} \\
        D\, \p_{n} u = \nu v-\mu u & \text{on $\{x_N=0\}\times{{\mathbb{R}}}^+$}\\
        -d\, \p_{n} v = \mu u-\nu v & \text{on $\{x_N=0\}\times{{\mathbb{R}}}^+$.}
    \end{array}\right.$$ As a first result, we obtain that arbitrarily small subsolutions of the desired type exist for all $0<c<c_g$.

\[0pr:1.3\] For every $0<c<c_g$, there exist $\e_0=\e_0(c)>0$ and $L=L(c)>0$ such that, for every $0<\e<\e_0$, problem admits a compactly supported subsolution of the form $\e(\un u,0)$, where $$\un u:=\begin{cases}
            e^{\frac{c}{2D}x_1}\psi_D(x) & \text{in $B_L$} \\
            0 & \text{otherwise,}
        \end{cases}$$ $B_L=B((0,\ldots,0,-(L+1)),L)$, and $\psi_D$ is the positive solution of the following eigenvalue problem $$\left\{\begin{array}{ll}
            -D\D\psi=\l_1(L)\psi & \text{in $B_L$} \\
            \psi=0 & \text{on $\p B_L$} \\
            \max_{B_L}\psi=1. &
        \end{array}\right.$$

Fix $c\in(0,c_g)$ and take $\t=\t(c)>0$ so that $$\label{0eq:1.14}
            g'(0)-\frac{c^2}{4D}-\t>0.$$ Since $\l_1$ decreases in $L$ and converges to $+\infty$ as $L\to 0$ and to $0$ as $L\to+\infty$, there exists a unique value of $L$ for which $\l_1$ equals the quantity in . For such an $L$, $\un u$ satisfies the linear equation $-D\D u+cu_{x_1}=(g'(0)-\t)u$ in $B_L$, thus thanks to the Fisher-KPP hypothesis , for $\e$ small enough, $\e\un u$ is a subsolution of the first equation in . It is then immediate to see that $\e(\un u,0)$ is a subsolution of the whole system.

By reasoning analogously in the upper half-space, we immediately obtain the following result.

\[0pr:1.4\] For every $0<c<c_f$, there exist $\e_0=\e_0(c)>0$ and $L=L(c)>0$ such that, for every $0<\e<\e_0$, problem admits a compactly supported subsolution of the form $\e(0,\un v)$, where $$\un v:=\begin{cases}
            e^{\frac{c}{2d}x_1}\psi_d(x) & \text{in $B_L$} \\
            0 & \text{otherwise,}
        \end{cases}$$ $B_L=B((0,\ldots,0,(L+1)),L)$, and $\psi_d$ is the positive solution of the following eigenvalue problem $$\left\{\begin{array}{ll}
            -d\D\psi=\l_1(L)\psi & \text{in $B_L$} \\
            \psi=0 & \text{on $\p B_L$} \\
            \max_{B_L}\psi=1. &
        \end{array}\right.$$

The last result is the construction of arbitrarily small, compactly supported, stationary subsolutions of for $c<c^*_\infty$, $c\sim c^*_\infty$, when the quantity $c^*_\infty$ defined in is given by $c_a$.

\[0pr:1.5\] Assume $$\label{0eq:1.14bis}
        \frac{D}{d}> 2-\frac{g'(0)}{f'(0)} \qquad \text{and} \qquad \frac{d}{D}> 2-\frac{f'(0)}{g'(0)}.$$ Then, for $c<c^*_\infty=c_a$, $c\sim c^*_\infty$, there exist arbitrarily small, non-negative, compactly supported, stationary subsolutions of .

Since the proof is quite long and requires many constructions, we divide it in several steps: in Step 1 we consider a penalized truncated version of the linearizion of around $(0,0)$, and we introduce and study some auxiliary curves associated to it; in Step 2 we define, using the curves of Step 1, an approximate speed of propagation $c^*_L$ and construct compactly supported subsolutions for $c<c^*_L$, $c\sim c^*_L$; finally, in Step 3, we show that, as $L\to+\infty$, $c^*_L$ converges to $c_a$. This, together with Step 2, will imply, by taking a sufficiently large $L$, the existence of the desired subsolutions for $c$ smaller and arbitrarily close to $c_a$.

*Step 1.* We first look for stationary solutions of a penalized version of the linearizion of around $(0,0)$, i.e. solutions of with additional transport in the $x_1$ direction and $g(u)$ and $f(v)$ respectively replaced by $(g'(0)-\t)u$ and $(f'(0)-\t)v$, $\t\sim 0$.

In addition, since we will look for compactly supported subsolutions, we require $u$ and $v$ to vanish at points $\{(x_2,\ldots,x_N): \left\lVert(x_2,\ldots,x_{N-1})\right\rVert=L,\, |x_N|=L\}$, where $L$ is a large number to be fixed later, as explained at the beginning of this proof, and the condition on the euclidean norm $\left\lVert(x_2,\ldots,x_{N-1})\right\rVert$ has to be taken into account only if $N\geq 3$. In analogy with the guess , we seek solutions having an exponential profile in the $x_1$ variable, while the vanishing condition leads us to consider three types of profile in the variable $x_N$: $$\label{0eq:1.15}
            (\un{u}_i,\un{v}_i)= e^{\a x_1}\psi(x_2,\ldots,x_{N-1})\left(\phi_{u,i}(x_N),\g\phi_{v,i}(x_N)\right), \qquad i\in\{1,2,3\},$$ where $\a$ and $\g$ are positive constants, $\psi$ (which is present only if $N\geq 3$) is the positive eigenfunction of problem $$\left\{
        \begin{array}{ll}
        -\D\psi=\l^2(L)\psi & \text{in $B:=B_{N-2}(0,L)$} \\
        \psi=0 & \text{on $\p B$,} 
        \end{array} \right.$$ normalized so that $\psi(0)=1$, $B_{N-2}(0,L)$ is the ball of radius $L$ centered at the origin of ${{\mathbb{R}}}^{N-2}$, and $$\begin{array}{rll}
        (\phi_{u,1},\phi_{v,1})\!\!\!\!&=(\sin(\b(x_N+L)),\sin(\d(L-x_N))), & \text{with } \b,\d\in\left(0,\frac{\pi}{L}\right), \\
        (\phi_{u,2},\phi_{v,2})\!\!\!\!&=(\sin(\b(x_N+L)),\sinh(\d(L-x_N))), & \text{with } \b\in\left(0,\frac{\pi}{L}\right), \quad \d>0, \\
        (\phi_{u,3},\phi_{v,3})\!\!\!\!&=(\sinh(\b(x_N+L)),\sin(\d(L-x_N))), & \text{with } \d\in\left(0,\frac{\pi}{L}\right), \quad \b>0.
        \end{array}$$ By plugging for $i=1$ into the penalized version of , we obtain the following system for the unknowns $\a,\b,\g,\d$: $$\label{0eq:1.16zero}
            \left\{ \begin{array}{l}
                c\a-D\a^2+D\l^2(L)+D\b^2=g'(0)-\t \\
                c\a-d\a^2+d\l^2(L)+d\d^2=f'(0)-\t \\
                D\b\cos(\b L)=\nu\g\sin(\d L)-\mu\sin(\b L) \\
                d\g\d\cos(\d L)=\mu\sin(\b L)-\nu\g\sin(\d L).
                \end{array} \right.$$ From the last equation of we find $$\g=\frac{\mu\sin(\b L)}{\nu\sin(\d L)+d\d\cos(\d L)},$$ and, in order $\g$ to be positive, we take $0<\d<\ov \d$, where $\ov \d\in\left(\frac{\pi}{2L},\frac{\pi}{L}\right)$ is the first value of $\d$ for which the denominator vanishes. Then, we substitute such a value of $\g$ in the third equation of , obtaining $$\label{0eq:1.16}
        \chi_{u,1}(\b):=D\b\cot(\b L)=\frac{-\mu d\d\cos(\d L)}{\nu\sin(\d L)+d\d\cos(\d L)}=:\chi_{v,1}(\d).$$ The function $\chi_{u,1}$ is analytic and decreasing in $\left(0,\frac{\pi}{L}\right)$, positive in $\left(0,\frac{\pi}{2L}\right)$, negative in $\left(\frac{\pi}{2L},\frac{\pi}{L}\right)$ and satisfies $$\label{0eq:1.16bis}
    \lim_{\b\da 0}\chi_{u,1}(\b)=\frac{D}{L}, \qquad \lim_{\b\da 0}\chi_{u,1}'(\b)=0, \qquad \lim_{\b\ua\frac{\pi}{L}}\chi_{u,1}(\b)=-\infty.$$ On the other hand, $\chi_{v,1}$ is analytic and increasing in $\left(0,\ov\d\right)$, negative in $\left(0,\frac{\pi}{2L}\right)$, positive in $\left(\frac{\pi}{2L},\frac{\pi}{L}\right)$ and satisfies $$\lim_{\d\da 0}\chi_{v,1}'(\d)=0, \qquad \lim_{\d\ua\ov\d}\chi_{v,1}(\d)=+\infty.$$ After all the considerations, if we denote by $\un\b\in\left(\frac{\pi}{2L},\frac{\pi}{L}\right)$ and $\un\d\in\left(\frac{\pi}{2L},\ov\d\right)$ the unique values of $\b$ and $\d$ satisfying, respectively, $\chi_{u,1}(\un\b)=\lim_{\d\da 0}\chi_{v,1}(\d)$ and $\chi_{v,1}(\un{\d})=\lim_{\b\da 0}\chi_{u,1}(\b)$, the implicit function theorem gives the existence of a 1-1 function $\d_1(\b):(0,\un\b)\to(0,\un{\d})$ such that relation holds true if and only if $\d=\d_1(\b)$. Such a function is analytic in $(0,\un \b)$, decreasing and satisfies $\lim_{\b\da 0}\d_1'(0)=0$.

In a similar way, if we pass to the case $i=2$, we obtain the system $$\label{0eq:1.17zero}
        \left\{ \begin{array}{l}
            c\a-D\a^2+D\l^2(L)+D\b^2=g'(0)-\t \\
            c\a-d\a^2+d\l^2(L)-d\d^2=f'(0)-\t \\
            D\b\cos(\b L)=\nu\g\sinh(\d L)-\mu\sin(\b L) \\
            d\g\d\cosh(\d L)=\mu\sin(\b L)-\nu\g\sinh(\d L),
        \end{array} \right.$$ whose the last equation gives $$\g=\frac{\mu\sin(\b L)}{\nu\sinh(\d L)+d\d\cosh(\d L)},$$ and no restriction on $\d$ is necessary now. By substituting such an expression into the third equation, we obtain $$\label{0eq:1.17}
        \chi_{u,2}(\b):=D\b\cot(\b L)=\frac{-\mu d\d\cosh(\d L)}{\nu\sinh(\d L)+d\d\cosh(\d L)}=:\chi_{v,2}(\d).$$ The function $\chi_{u,1}$ coincides with $\chi_{u,2}$, while $\chi_{v,2}$ is analytic, negative, decreasing and satisfies $$\lim_{\d\da 0}\chi_{v,2}(\d)=\lim_{\d\da 0}\chi_{v,1}(\d)>\lim_{\d\to +\infty}\chi_{v,2}(\d)>-\infty, \qquad \lim_{\d\da 0}\chi_{v,2}'(\d)=0.$$ Thus, if $\un\b$ is as above, and we denote by $\ov\b\in\left(\un\b,\frac{\pi}{L}\right)$ the unique value of $\b$ such that $\chi_{u,2}(\b)=\lim_{\d\to+\infty}\chi_{v,2}(\d)$, there exists a 1-1 analytic function $\d_2(\b):(\un\b,\ov\b)\to(0,+\infty)$ such that relation holds true if and only if $\d=\d_2(\b)$. Such a function is increasing and satisfies $\lim_{\b\ua\ov\b}\d_2'(\b)=+\infty$.

Finally, for $i=3$, the system obtained by plugging into the linearization reads $$\label{0eq:1.18zero}
            \left\{ \begin{array}{l}
                c\a-D\a^2+D\l^2(L)-D\b^2=g'(0)-\t \\
                c\a-d\a^2+d\l^2(L)+d\d^2=f'(0)-\t \\
                D\b\cosh(\b L)=\nu\g\sin(\d L)-\mu\sinh(\b L) \\
                d\g\d\cos(\d L)=\mu\sinh(\b L)-\nu\g\sin(\d L),
            \end{array} \right.$$ whose last equation gives $$\g=\frac{\mu\sinh(\b L)}{\nu\sin(\d L)+d\d\cos(\d L)},$$ which is positive for $0<\d<\ov\d$, and, once substituted in the the third equation, yields to $$\label{0eq:1.18}
            \chi_{u,3}(\b):=D\b\coth(\b L)=\frac{-\mu d\d\cos(\d L)}{\nu\sin(\d L)+d\d\cos(\d L)}=:\chi_{v,3}(\d).$$ The function $\chi_{v,3}$ coincides with $\chi_{v,1}$, while $\chi_{u,3}$ is analytic, positive and increasing in $\left(0,+\infty\right)$, and it satisfies $$\label{0eq:1.18bis}
        \lim_{\b\da 0}\chi_{u,3}(\b)=\frac{D}{L}, \qquad \lim_{\b\da 0}\chi_{u,3}'(\b)=0, \qquad \lim_{\b\to+\infty}\chi_{u,3}(\b)=+\infty.$$ Thus, recalling that we have set $\un\d\in\left(\frac{\pi}{2L},\ov\d\right)$ to be the value satisfying $\chi_{v,3}(\un\d)=\frac{D}{L}=\lim_{\b\da 0}\chi_{u,3}(\b)$, there exists a 1-1 increasing analytic function $\d_3(\b):(0,+\infty)\to(\un\d,\ov\d)$ such that relation holds true if and only if $\d=\d_3(\b)$. In addition, $\d_3$ satisfies $\lim_{\b\da 0}\d_3'(0)=0$.

Now we glue the functions defined above as follows $$\label{0eq:1.19}
        \mathfrak{d}(\b):=\begin{cases}
        \d_1^2(\b) & \text{for $\b\in(0,\un\b)$,} \\
        -\d_2^2(\b) & \text{for $\b\in(\un\b,\ov\b)$,} \\
        \d_3^2(-\b) & \text{for $\b\in(-\infty,0)$,}
        \end{cases}$$ and we claim that $\mathfrak{d}\in{\mathcal}{C}^1(-\infty,\ov\b)$ and that it is analytic in $(0,\ov\b)$. For the latter point, since we already know the analyticity of $\d_1$ and $\d_2$ in their respective domains, it remains to check it at the gluing point. To this end, consider the extended function $\d_1:(0,\ov\b)\to{{\mathbb{C}}}$ obtained by applying the complex version of the implicit function theorem to relation . As $\cosh(i x)=\cos(x)$ and $\sinh(i x)=i\sin(x)$ for every $x\in{{\mathbb{C}}}$, by comparing and , the uniqueness given by the implicit function theorem entails that, for $\b\in(\un\b,\ov\b)$, $\d_2(\b)=i\d_1(\b)$. Thus $\d_2^2(\b)=-\d_1^2(\b)$, and we obtain that, by construction, $\mathfrak{d}(\b)$ is analytic at $\b=\un\b$. Passing to the continuity and differentiability in $0$, such a property follows from the definition of $\d_1$ and $\d_3$, together with the first two relations in and .

With this definition of $\mathfrak{d}(\b)$ and recalling the monotonicities of the functions $\d_i$ discussed above, we have that $\mathfrak{d}(\b)$ is decreasing and bounded from above. Thus, if we set $$\label{0eq:1.19bis}
\un c:=\inf\{c>0: \exists! \, \b\in(-\infty,\ov\b) \text{ s.t. } c^2=4d\left(f'(0)-\t-d\l^2(L)-d\mathfrak{d}(\b)\right)\},$$ for every $c>\un c$ we have that the second equation of each of the systems , and can be solved for $\a$ as a function of $\b$ as follows $$\label{0eq:1.19ter}
        \a_d^{\pm}(c,\b,L):=\frac{c\pm\sqrt{c^2-4d\left[f'(0)-\t-d\l^2(L)-d\mathfrak{d}(\b)\right]}}{2d},$$ with $\b$ lying in the corresponding domain as in and, additionally, $\b\in(-\infty,\breve{\b}(c))$, where $\breve{\b}(c)\in(-\infty,\ov\b)$ is the unique value of $\b$ for which the argument of the square root in vanishes. Moreover, $\b\mapsto\a_d^{-}(c,\b,L)$ is increasing in its domain, while $\b\mapsto\a_d^{+}(c,\b,L)$ is decreasing. Gluing together these curves gives a differentiable curve, namely $$\Si_d(c,L):=\left\{\left(\b,\a_d^{\pm}(c,\b,L)\right):\b\in\left(-\infty,\breve{\b}(c)\right)\right\}.$$

Turning now to the first equation of systems , and , we can handle all of them together by setting, when the quantities appearing are defined, $$\label{0eq:1.20}
        \a_D^{\pm}(c,\b,L):=\begin{cases}
        \frac{c\pm\sqrt{c^2-4D\left[g'(0)-\t-D\l^2(L)+D\b^2\right]}}{2D} & \text{for $\b<0$} \\
        \frac{c\pm\sqrt{c^2-4D\left[g'(0)-\t-D\l^2(L)-D\b^2\right]}}{2d} & \text{for $\b\geq 0$.}
        \end{cases}$$ In order to study in detail the domain of definition of such a function, if $0<c<2\sqrt{D\left[g'(0)-\t-D\l^2(L)\right]}$, we denote by $\hat{\b}(c)$ the unique positive value of $\b$ such that $4D^2\b^2=4D(g'(0)-\t-D\l^2(L))-c^2$, while, if $c\geq 2\sqrt{D(g'(0)-\t-D\l^2(L))}$, we define $\hat{\b}(c)$, as the unique negative value of $\b$ such that $4D^2\b^2=c^2-4D(g'(0)-\t-D\l^2(L))$. With this notation, $\a_D^{\pm}(c,\b,L)$ is defined for $\b\geq\hat{\b}(c)$, and we consider in the upper half-plane $(\b,\a)$ the curve $$\Si_D(c,L):=\left\{(\b,\a_D^{\pm}(c,\b,L)):\b\geq\hat{\b}(c)\right\}.$$ The right part of $\Si_D$, corresponding to the case $\b\geq 0$ in , consists of the branches of an hyperbola, while the left one, corresponding to the case $\b<0$, consists, when it is defined, of a half-circle with center $\left(0,\frac{c}{2D}\right)$ and radius $\frac{\sqrt{c^2-4D\left[g'(0)-\t-D\l^2(L)\right]}}{2D}$. In addition, it is easy to see that, even when $\Si_D$ is defined both for $\b<0$ and $\b\geq 0$, such a curve is differentiable.

*Step 2.* We now analyze the behavior with respect to $c$ of the curves that we have introduced in the previous step. First of all, since $\d_3(\b)\in(\un\d,\ov\d)\subset\left(\frac{\pi}{2L},\frac{\pi}{L}\right)$, we can fix $L$ sufficiently large so that $\d^2(\b)$ is arbitrarily close to $0$ uniformly for $\b\in(-\infty,0)$ and, as a consequence, so that the quantity $\un c$ introduced in is strictly positive. In such a case, as $c\da\un c$, $\breve{\b}(c)\to-\infty$, while $\hat{\b}(c)\to\hat{\b}(\un c)$, entailing that $\Si_d(c,L)$ and $\Si_D(c,L)$ do not intersect for $c>\un c$, $c\sim \un c$.

In addition, we observe that $c\mapsto\a_d^{+}(c,\b,L)$ is increasing, while $c\mapsto\a_d^{-}(c,\b,L)$ is decreasing whenever $\a_d^{-}(c,\b,L)>0$, which is the case that we consider, since we look for positive $\a$'s. The same monotonicities hold true for $\a_D^\pm$.

Finally, as $c\to+\infty$, $\hat{\b}(c)\to-\infty$, $\breve{\b}(c)\to\ov\b$, and the curves $\a_d^{-}$ and $\a_D^{-}$ converge to $0$, while $\a_d^{+}$ and $\a_D^{+}$ to $+\infty$, always in their domain of definition.

As a consequence, all these considerations and the differentiability of the curves $\Si_d$ and $\Si_D$ imply that there is a smallest value of $c$ (greater than $\un c$), denoted by $c^*_L$, for which such curves intersect, being tangent.

By construction, the curves $\Si_d(c,L)$ and $\Si_D(c,L)$ have no real intersections for $c<c^*_L$, $c\sim c^*_L$. Nonetheless, if the tangency point for $c=c_L^*$, which will be denoted by $(\b^*_L,\a^*_L)$, satisfies $\b^*_L\neq 0$, i.e. it lies in the region in which the curves are analytic, by using Rouché's theorem as in @RTV [Theorem A.1], it is possible to find complex intersections for $c<c^*_L$, $c\sim c^*_L$, giving rise to a complex $(\un u,\un v)$. The real part of $(\un u,\un v)$, which is still a solution of the linearized system, oscillates in $x_1$, but, by taking one positive bump, extending to $(0,0)$ in the complement, and multiplying by a small positive constant, we obtain a nonnegative, compactly supported subsolution of .

If $\b^*_L=0$, instead, we can consider the curves only for $\b\geq 0$, extend them analytically for $\b<0$, $\b\sim 0$ and apply now Rouché's theorem to such analytic continuation to get compactly supported subsolutions of .

*Step 3.* We finally show that $c^*_L$ converges to $c_a$ as $L\to+\infty$. For convenience we prove it for $\t=0$ and, since all the constructions performed here depend in a differentiable way on $\t$, the same will hold true for $\t\sim 0$.

To achieve our purpose, we need to study the behavior of $\Si_D(c,L)$ and $\Si_d(c,L)$ as $L\to+\infty$. As already shown, when $L\to+\infty$, $\d^2(\b)\to 0$ uniformly for $\{\b<0\}$, and $\l(L)\to 0$. Recall that $\b\to\d^2(\b)$ is increasing; thus, if $c<c_f$ and $L$ is large, we have $c^2-4d(f'(0)-d\l^2(L)-d\d^2(\b))<0$ for all $\b<\ov\b$, entailing that $\Si_d(c,L)$ is not defined. Thus, we consider $c\geq c_f$. In such a case, since $c^2-4d(f'(0)-d\l^2(L)-d\d^2(\b))$ is positive for $\b=0$ and $L$ large, we have $\breve{\b}(c)>0$. Moreover, $\breve{\b}(c)<\ov\b<\frac{\pi}{L}$ thus $\breve{\b}(c)\to 0$ as $L\to+\infty$. This, together with the monotonicity of $\b\to\d^2(\b)$, shows that the curve $\Si_d(c,L)$ converges in the Hausdorff distance, for all $c\geq c_f$, to the following set $$\label{0eq:1.30}
        \Si_d(c,+\infty):=\left\{\left(\b,r_d^\pm(c)\right):\b<0\right\}\bigcup\left\{(0,\a):r_d^-(c)\leq\a\leq r_d^+(c)\right\}.$$

The behavior of $\Si_D(c,L)$ is simpler, since, by continuity, it converges to the curve which is just obtained by neglecting the term $D\l^2(L)$ in and, as a consequence, is of the same type as $\Si_D(c,L)$.

Now we prove that $c^*_L<c_a$ for sufficiently large $L$. Indeed, recall from Proposition \[0pr:1.2bis\] that $c_a>\max\{c_f,c_g\}$, and, as shown above, this implies $\breve{\b}(c_a)>0$ and $\hat\b(c_a)<0$ for large $L$. Moreover, from the definition of $c_a$ (see again Proposition \[0pr:1.2bis\]), we have $r_d^+(c_a)=r_D^-(c_a)$ or $r_d^-(c_a)=r_D^+(c_a)$. Let us treat the first case, the other one being analogous. The expression of the curves gives $$\a_d^+(c_a,0,L)>r_d^+(c_a)=r_D^-(c_a)>\a_D^-(c_a,0,L),$$ and, since the left and the right quantities converge to the central ones for $L\to+\infty$, we have that $\Si_d(c_a,L)$ lies inside the circle forming the left part of $\Si_D(c_a,L)$ for $\b\sim 0$, while it lies outside for very negative $\b$, entailing that the curves intersect and, by definition, $c^*_L<c_a$.

In order to conclude and prove the desired limit, we distinguish three cases, as we did in the proof of Proposition \[0pr:1.2bis\], according to the relative position of $c_f$ and $c_g$.

If $c_f<c_g$, then, for $c=c_g$, $\Si_D(c,L)$ converges, as $L\to+\infty$, to the degenerate hyperbola $\{(\b,\frac{c_g}{2D}\pm\b),\b\geq 0\}$, which, thanks to the assumption , does not intersect the limiting curve . By continuity, the curves $\Si_D(c_g,L)$ and $\Si_d(c_g,L)$ do not intersect for large $L$, implying $\liminf_{L\to+\infty}c^*_L>c_g$. If, by contradiction, $\liminf_{L\to+\infty}c^*_L<c_a$, a similar reasoning shows that the curves $\Si_D(c,L)$ and $\Si_d(c,L)$ do not intersect for large $L$, against the definition of $c^*_L$.

The other cases can be handled similarly, thus we do not provide the details.

Once that we have constructed, on the one hand, supersolutions consisting in a fixed profile that decays to $0$ at infinity in the $x_1$ direction, and moving with speed $c\geq c^*_\infty$ and, on the other hand, stationary subsolutions with compact support for the problem with transport term with speed $c<c^*_\infty$, $c\sim c^*_\infty$, again in the $x_1$ direction, we are able to prove that $c^*_\infty$ is the asymptotic speed of propagation for problem in the direction $x_1$. As a consequence of the invariance of the problem by rotations around the axis $\{x_1=\ldots=x_{N-1}=0\}$, we will immediately obtain that it gives the value of the asymptotic speed of propagation in every direction $x_i$, $1\leq i\leq N-1$.

\[0th:1.4\] Assume and and recall the quantity $c^*_\infty$ introduced in . Then, for any $i\in\{1,\ldots,N-1\}$ and every non-negative, continuous, bounded, compactly supported initial datum $(u_0,v_0)$, the solution $(u,v)$ of satisfies:

(i) \[0th:1.4.i\] for all $c>c^*_\infty$, $\lim_{t\to\infty}\sup_{|x_i|\geq ct}(u,v)=(0,0)$, uniformly in $x_j$, for all $j\neq i$,

(ii) \[0th:1.4.ii\] for all $0<c<c^*_\infty$, $\lim_{t\to\infty}\sup_{|x_i|\leq ct}|(u,v)-(U,V)|=0$, locally uniformly in $x_j$, for all $j\neq i$, where $(U,V)$ is the unique steady state of problem given by Proposition \[0th:1.3\].

As remarked above, using the symmetry of the problem, we can assume without loss of generality that $i=1$. For part , we consider the supersolution $(\ov u,\ov v)$ of the form with the $-$sign and $c=c^*_\infty$. If we multiply it by a positive constant $k$, we still have a supersolution of the linearized problem , thus a supersolution of . As a consequence, since the initial datum is bounded, $k(\ov u,\ov v)$ lies, for large $k$, above $(u_0,v_0)$. Take now $c>c^*_\infty$ and $x_1\geq ct$; the comparison principle gives $$\label{0eq:1.31}
            (u,v)\leq ke^{-\a(x_1-c^*_\infty t)}\left(1,\frac{\mu}{\nu}\right)\leq  ke^{-\a(c-c^*_\infty) t}\left(1,\frac{\mu}{\nu}\right)\to (0,0) \quad \text{as $t\to+\infty$.}$$ The result for $x_1\leq -ct$ follows either by taking into account the symmetry of the problem with respect to the change of variable $x_1\mapsto -x_1$ or by taking the supersolution with the $+$sign in .

The uniform convergence with respect to the variables $x_j$, $j\neq i$, follows by observing that we can take the supremum in each $x_j$ in before passing to the limit in $t$.

For part , take $c<c^*_\infty$, $c\sim c^*_\infty$ and consider $$(\tilde u,\tilde v)(x_1,\ldots,x_N,t):=(u,v)(x_1+ct,x_2,\ldots,x_N,t).$$ In order to conclude by applying @BRR2 [Lemma 4.1] (see also @RTV [Lemma 4.4] for a proof of it), it suffices to show that $(\tilde u,\tilde v)$ converges to $(U,V)$ as $t\to+\infty$. First of all, we observe that it solves and has $(u_0,v_0)$ as an initial datum. Thus, thanks to the strong comparison principle, it is strictly positive at any positive time. As a consequence, at time $t=1$, we can place below such a solution the arbitrarily small stationary subsolutions of constructed either in Proposition \[0pr:1.3\] or in Proposition \[0pr:1.4\] or in Proposition \[0pr:1.5\]. By reasoning as in Proposition \[0pr:1.1\] and Proposition \[0pr:1.2\], we can prove that the solution of the parabolic problem starting from the subsolution converges, locally uniformly, to $(U,V)$. Therefore, the same occurs to $(\tilde u,\tilde v)$, as we wanted.

We conclude this section with a diagram (see Figure \[0fig:1.1\]) that summarizes the possible values of $c^*_\infty$ according to the different ranges of the diffusion and reaction parameters ($D,d,g'(0),f'(0)$) appearing in the system, and we prove the following qualitative properties of $c^*_\infty$.

![The value of the asymptotic speed of propagation $c^*_\infty$ for problem , as defined in : in red, below the line $\frac{D}{d}= 2-\frac{g'(0)}{f'(0)}$, the region of the parameters for which $c^*_\infty=c_f$ and in blue, above the upper hyperbola $\frac{d}{D}= 2-\frac{f'(0)}{g'(0)}$, which is obtained from the previous line by interchanging the diffusion parameters and the derivatives at $0$ of the reaction terms, the region where $c^*_\infty=c_g$. In between, the region (containing the purple hyperbola $\frac{D}{d}=\frac{f'(0)}{g'(0)}$, corresponding to $c_f=c_g$) where $c^*_\infty$ equals the anomalous speed $c_a$ introduced in . The dotted regions correspond to the cases in which the last inequality in is strict.<span data-label="0fig:1.1"></span>](Rinfinity2.pdf)

\[0pr:1.6\] The asymptotic speed of propagation $c^*_\infty$ of problem satisfies the following properties:

(i) \[0pr:1.6.0\] the following inequalities hold true $$\label{1eq:1.12}
                \max\{c_f,c_g\}\leq c^*_\infty\leq 2\sqrt{\max\{D,d\}\max\{g'(0),f'(0)\}},$$ and the last inequality it is strict if and only if either $D>d$ and $f'(0)>g'(0)$, or $D<d$ and $f'(0)<g'(0)$;

(ii) \[0pr:1.6.i\] the function $c^*_\infty(d,D,f'(0),g'(0))$ is continuous;

(iii) \[0pr:1.6.ii\] the map $D\mapsto c^*_\infty(D)$ is non-decreasing;

(iv) \[0pr:1.6.iii\] $\lim_{D\to+\infty}\frac{c^*_\infty(D)}{\sqrt{D}}\in(0,+\infty)$.

\(i) The first inequality follows immediately from the construction of $c^*_\infty$ performed in Proposition \[0pr:1.2bis\].

Regarding the second one, up to interchanging the role of the two half-spaces, it will suffice to treat the case $D>d$. The inequality is trivial if $c^*_\infty=c_f$, since in this case, as it can be seen from Figure \[0fig:1.1\], necessarily $f'(0)>g'(0)$, thus $$c^*_\infty=c_f<2\sqrt{Df'(0)}=2\sqrt{\max\{D,d\}\max\{g'(0),f'(0)\}}.$$ If $c^*_\infty=c_g$ and $g'(0)\geq f'(0)$, then $$c^*_\infty=c_g=2\sqrt{Dg'(0)}=2\sqrt{\max\{D,d\}\max\{g'(0),f'(0)\}},$$ thus equality holds true. Moreover, if $c^*_\infty=c_g$ and $g'(0)<f'(0)$, then $$c^*_\infty=c_g=2\sqrt{Dg'(0)}<2\sqrt{Df'(0)}=2\sqrt{\max\{D,d\}\max\{g'(0),f'(0)\}},$$ as desired. Finally, when $c^*_\infty=c_a$, recalling the expression of $c_a$ (see ), our goal is to show that $$\frac{|Df'(0)-dg'(0)|}{\sqrt{(D-d)(f'(0)-g'(0))}}<2\sqrt{Df'(0)},$$ which is equivalent to $$y^2-y(6x-4x^2)+4x-3x^2<0, \qquad \text{ where we have set } x:=\frac{D}{d}, \quad y:=\frac{g'(0)}{f'(0)}.$$ Direct computations show that, if $x>1$ and $0<y<1$, the region in the diagram satisfying such a relation contains the one that we are considering, which is $2-x<y<\frac{x}{2x-1}$.

By analyzing all the previously treated cases, we also immediately see when the inequality is strict or large, obtaining the desired result.

\(ii) It is obvious that $c^*_\infty$ is continuous in the interior of the regions where it coincides with $c_f$, $c_g$ or $c_a$, thus we only have to prove the continuity when we pass from one region to the other. Direct computations show that relation $c_a>c_g$ can be equivalently rewritten as $$(y(2x-1)-x)^2>0, \qquad \text{ where } x:=\frac{D}{d}, \quad y:=\frac{g'(0)}{f'(0)},$$ thus we see that, as $y$ converges to the hyperbola $\frac{x}{2x-1}$, which is the one that separates the two regimes, the value of $c_a$ converges to $c_g$. (Incidentally, this also shows that indeed $c_a>c_g$ in the region where $c^*_\infty=c_a$). Similarly, $c_a>c_f$ is equivalent to $(y-(2-x))^2>0$, thus $c_a$ converges to $c_f$ as $y$ approaches the line $2-x$.

\(iii) Once that we know the continuity from the previous point, in order to obtain the monotonicity of $c^*_\infty(D)$ is suffices to prove the monotonicity in each region separately. Once again, the result is direct when $c^*_\infty=c_f$ or $c^*_\infty=c_g$. Let us focus on the case when $c^*_\infty=c_a$ for $D>d$ and $f'(0)<g'(0)$ (the other case can be treated analogously): we have $$c^{*'}_\infty(D)=\left(\frac{Df'(0)-dg'(0)}{\sqrt{(D-d)(f-g)}}\right)'=\frac{Df'(0)-2df'(0)+dg'(0)}{2(D-d)^{3/2}(f'(0)-g'(0))^{1/2}},$$ and the numerator in the last expression is positive, since we are in the region where $\frac{D}{d}>2-\frac{g'(0)}{f'(0)}$.

\(iv) This property follows immediately by analyzing the expressions of $c_g$ and $c_a$.

It is not possible to expect, in general, higher regularity than the one stated in Proposition \[0pr:1.6\], since, if we consider $g'(0)=f'(0)$, we observe that $c^*_\infty(D)$ is not differentiable at $D=d$, since it passes from being constant to take the value $2\sqrt{Dg'(0)}$.

Cylinder and its complement with Fisher-KPP nonlinearities {#section3}
==========================================================

We pass now to problem , i.e. we consider a cylinder with Fisher-KPP nonlinearities both in the interior and its complement. For this geometry, it will be convenient to denote $x:=x_1$ and $y:=(x_2,\ldots,x_N)$.

As in the previous section, we will first determine the long-time behavior for system , while, in the second part, we characterize its asymptotic speed of propagation in the $x$ direction, as well as some qualitative properties of such a speed.

Long-time behavior for system  {#1section2}
------------------------------

The first result that we obtain is the counterpart of Proposition \[0pr:1.1\] for this new geometry.

\[1pr:2.1\] Assume and . Then, for every continuous, bounded and non-negative $(u_0,v_0)$, there exist two positive bounded stationary solutions $(U_i,V_i)$ of , $i\in\{1,2\}$, which are $x$-independent, invariant by rotations around the axis $\{y=0\}$, and such that the solution of starting from $(u_0,v_0)$ satisfies $$    (U_1,V_1)\leq\liminf_{t\to+\infty}(u,v)\leq\limsup_{t\to+\infty}(u,v)\leq(U_2,V_2)$$ locally uniformly in each domain.

The same supersolution defined in can be used here to show the existence of $(U_2,V_2)$ and its $x$-invariance as in the proof of Proposition \[0pr:1.1\]. In addition, the axial symmetry of the supersolution is inherited by the steady state.

As for the stationary subsolution that allows us to construct $(U_1,V_1)$, we have to reason a little differently, since now $\O$ has a bounded section. In this case, we consider the eigenvalue problem $$\label{1eq:2.2}
        \left\{
        \begin{array}{ll}
        -\D_y \phi=\l \phi & \text{ in $A_L:=A(R+1,L)$} \\
        \phi=0 & \text{ on $\p A_L$,}
        \end{array}\right.$$ where $\D_y$ denotes the Laplacian with respect to the $y$ variables and $A(R+1,L)$ the annulus of points in ${{\mathbb{R}}}^{N-1}$ such that $R+1<|y|<L$. If $\l_1=\l_1(L)$ is the principal eigenvalue of and $\phi_1$ the associated positive eigenfunction, normalized so that $\max_{A_L}\phi_1=1$, we fix $\a:=\sqrt{\frac{f'(0)}{2d}}$, and we take a sufficiently large $L$ so that $d\a^2+d\l_1(L)<f'(0)$, then there exists $\e_0>0$ such that, setting $$        \un v:=\left\{
        \begin{array}{ll}
        \cos(\a x)\phi_1(y) & \text{ for $x\in\left(\frac{-\pi}{2\a},\frac{\pi}{2\a}\right)$, $y\in A_L$} \\
        0 & \text{ otherwise,}
        \end{array}\right.$$ $\e(0,\un v)$ is a strict non-negative stationary subsolution of for all $\e\in(0,\e_0)$ which is, in addition, axially symmetric. We then conclude the proof by reasoning as in Proposition \[0pr:1.1\].

The next question is the uniqueness of the steady state with the properties described in the previous proposition. Numerical observations that we have performed show that, without adding additional hypothesis on the nonlinearities, uniqueness does not hold in general. For this reason we assume the strong KPP conditions and . Moreover, we provide a different proof than the one for the analogous result in Section \[0section1\], which is based on PDE techniques and is inspired by @BHR.

\[1pr:2.2\] Assume , , and . Then, there exists a unique positive, bounded and $x$-independent and axially symmetric steady state for problem , which will be denoted by $(U,V)$.

The existence of a steady state with the properties of the statement follows from Proposition \[1pr:2.1\]. Let now $(U,V)$ be any such a steady state. As a first step, we prove that $$        \inf_{\ov\O} U>0 \quad \text{and} \quad \inf_{{{\mathbb{R}}}^N\setminus\O} V>0.$$ Since $(U,V)$ is positive and $x$-independent, it is excluded that it tends to $0$ as $x\to\pm\infty$. Moreover, the strong positivity property excludes the existence of an interior point in which either $U$ or $V$ vanish. In addition, the fact that neither of them approaches $0$ at a point $(x_0,y_0)\in\p\O$ can be excluded using the Hopf boundary lemma. Indeed, if for example $U(x_0,y_0)=0$, then $$\label{1eq:2.4bis}
        0> D\p_nU(x_0,y_0)=\nu V(x_0,y_0)\geq 0,$$ a contradiction. The case $V(x_0,y_0)=0$ can be excluded analogously. Finally, using the rotational symmetry of $(U,V)$ and reasoning on its profile, it is possible to show as in Proposition \[0pr:1.2\] that $V(y)$ converges to $S>0$ as $|y|\to+\infty$.

Assume now that there exist two steady states $(U_i,V_i)$, $i\in\{1,2\}$, with the above-mentioned properties. Since $(U_1,V_1)$ is positive and bounded, for $\g=0$ we have $\g(U_2,V_2)<(U_1,V_1)$, while $\g(U_2,V_2)>(U_1,V_1)$ for large $\g$. By continuity, if we set $$\g^*:=\sup\left\{\g>0: \g(U_2,V_2)<(U_1,V_1) \right\},$$ which is positive and finite, we have $U_1-\g^*U_2\geq 0$, $V_1-\g^*V_2\geq 0$, and $$\label{1eq:2.5}
        \inf_{\ov\O} U_1-\g^*U_2=0 \quad \text{or} \quad \inf_{{{\mathbb{R}}}^N\setminus\O} V_1-\g^*V_2=0.$$ Let us show that $\g^*\geq 1$, which will entail that $(U_1,V_1)\geq(U_2,V_2)$. By exchanging the roles of $(U_1,V_1)$ and $(U_2,V_2)$ and repeating the same argument, we conclude that the two steady states coincide. Assume by contradiction $\g^*<1$ and consider the several possibilities given by . Suppose, for example, that $U_1$ and $\g^*U_2$ touch in a point $(x_0,y_0)\in\O$. Then, the function $Z_U:=U_1-\g^* U_2$ is non-negative, vanishes at $(x_0,y_0)$ (thus has an interior minimum), and, thanks to and the contradiction hypothesis $\g^*<1$, it satisfies $$\label{1eq:2.6}
        -D\D Z_U=g(U_1)-\g^*g(U_2)>g(U_1)-g(\g^* U_2)=\tilde g(z)Z_U \quad \text{ in $\O$},$$ where $\tilde g(z)$ is the bounded function defined as $\frac{g(U_1(z))-g(\g^* U_2(z))}{U_1(z)-\g^* U_2(z)}$ if the denominator does not vanish and, for example, as $1$ otherwise. The strong positivity property thus implies that $Z_U=0$ everywhere, which contradicts .

In the same way we exclude the possibility of a contact point between $V_1$ and $\g^* V_2$ in ${{\mathbb{R}}}^N\setminus\ov\O$, while the existence of a contact point between on $\p\O$ can be excluded as in .

Finally, if there existed a sequence $(y_n)_{n\in{{\mathbb{N}}}}$ such that $|y_n|\to\infty$ and $\inf_{n\in{{\mathbb{N}}}} (V_1(y_n)-\g^*V_2(y_n))=0$, then, thanks to elliptic estimates, the functions $$V_{n,1}(y):=V_1(y+y_n), \qquad  V_{n,2}(y):=V_2(y+y_n),$$ which are bounded and bounded away from 0, would respectively converge pointwise to some bounded, non-negative functions $V_{\infty,1}$ and $V_{\infty,2}$, which are defined in ${{\mathbb{R}}}^N$, satisfy $-D\D V_{\infty,i}=f(V_{\infty,i})$ for $i\in\{1,2\}$, and such that $Z_V:=V_{\infty,1}-\g^*V_{\infty,2}$ is non-negative and vanishes in $0$. By reasoning as in , we exclude also this last possibility.

An immediate consequence of Propositions \[1pr:2.1\] and \[1pr:2.2\], we obtain the proof of the invasion result in Theorem \[1th:3.3\].

Construction of the asymptotic speed of propagation for  {#1subsection:c^*_2}
--------------------------------------------------------

The general scheme of construction of the asymptotic speed of propagation for problem is the same as in Section \[0section1\]. Nonetheless, the different geometry entails some technical changes in the study of the linearization of around $(0,0)$, which we detail hereafter. Indeed, due to the symmetry of the problem, we now look for positive axially symmetric (super)solutions to such a system of the following type $$\label{1eq:3.2}
(\ov u,\ov v)=e^{\mp\a(x\mp ct)}|y|^{-\tau}\left(J_\tau(\b|y|),\g K_\tau(\d|y|)\right),$$ where $J_\tau$ is the Bessel function of first kind and $K_\tau$ the modified Bessel function of second kind of order $\tau=\frac{N-3}{2}$ (observe that $\tau$ is either integer or half-integer and $\tau\geq -1/2$), while $\a,\b,\g,\d$ are positive constants to be determined.

If we denote by $j_{\tau}$ the first positive zero of $J_\tau$, we will require $\b R<j_\tau$, in order $\ov u$ to be positive in $\ov\O$. Moreover, thanks to , $\ov u$ is well defined and regular in the whole domain. On the other hand, $K_\tau(r)$ is positive and defined for all $r>0$.

Since $J_\tau$ (resp. $K_\tau$) satisfies the Bessel equations (resp. ), we have that $e^{\pm\a(x\mp ct)}\ov u$ (resp. $e^{\pm\a(x\mp ct)}\ov v$) is a positive solution of $-\D_y\phi=\b^2\phi$ in $\{|y|<R\}$ (resp. of $\D_y\phi=\d^2\phi$ in $\{|y|>R\}$).

In addition, taking into account the relations involving their derivatives given in and , we have that looking for positive solutions (respectively supersolutions) of the form of the linearized problem around $(0,0)$ is equivalent to find positive solutions of the following system (respectively with the $\lq\lq="$ sign replaced by $\lq\lq\geq"$) which involves the parameters $\a,\b,\g,\d$: $$\label{1eq:3.5}
\left\{\begin{array}{l}
c\a-D\a^2+D\b^2=g'(0) \\
c\a-d\a^2-d\d^2=f'(0) \\
-D\b J_{\tau+1}(\b R)=\nu\g K_{\tau}(\d R)-\mu J_{\tau}(\b R) \\
d\g\d K_{\tau+1}(\d R)=\mu J_{\tau}(\b R)-\nu\g K_{\tau}(\d R).\\
\end{array}\right.$$ From the third equation we obtain $$\label{1eq:3.6}
\g=\frac{\mu J_{\tau}(\b R)-D\b J_{\tau+1}(\b R)}{\nu K_{\tau}(\d R)},$$ and, by plugging this expression in the fourth equation, we obtain $$\label{1eq:3.7}
\chi_v(\d)=:d\d \frac{K_{\tau+1}(\d R)}{K_{\tau}(\d R)}=\frac{\nu D\b J_{\tau+1}(\b R)}{\mu J_{\tau}(\b R)- D \b J_{\tau+1}(\b R)}:=\chi_u(\b),$$ which, by using the functions introduced in and , can be equivalently rewritten as $$\label{1eq:3.8}
\chi_v(\d)=\frac{d}{R}h_v(\d R)=\frac{\nu D h_u(\b R)}{\mu R - D h_u(\b R)}=\chi_u(\b).$$ From the expression of $\chi_u(\b)$ given in , if we denote by $$\label{1eq:3.ovb}
    \ov \b\!<\!\frac{j_\tau}{R} \text{ the first positive zero of the function } \b\mapsto\mu J_{\tau}(\b R)\!-\!D \b J_{\tau+1}(\b R),$$ which is positive for $\b\sim0$ and negative for $\b=j_\tau/R$ (see ), we have that $\g$ in is positive, and $\chi_u(\b)$ is positive and well defined for $\b\in[0,\ov \b)$.

Thanks to Lemma \[le:A.1\], $h_u$ is increasing in $(0,j_\tau)$, while gives that $h_u(0)=h'_u(0)=0$. Collecting all these properties, we have $$\chi_u(0)=\chi'_u(0)=0, \qquad \chi_u(\b)>0< \chi'_u(\b) \text{ for $\b\in(0,\ov\b)$}, \qquad \lim_{\b\ua\ov\b}\chi_u(\b)=+\infty.$$ On the other hand, using and , we obtain $$\label{1eq:3.9}
\lim_{\d\da 0}\chi_v(\d)=\frac{d}{R}\max\{0,N-3\}, \qquad \lim_{\d\da 0}\chi'_v(\d)=\begin{cases}
d & \text{if $N=2,4$} \\
+\infty & \text{if $N=3$} \\
0 & \text{otherwise.} \\
\end{cases}$$ Moreover, thanks to , we have that $\chi_v$ is increasing, while gives $$\lim_{\d\ua+\infty}\frac{\chi_v(\d)}{\d}=d.$$ Let us denote by $\un\b$ the unique value of $\b\in[0,\ov\b)$ such that $\lim_{\d\da0}\chi_v(\d)=\chi_u(\b)$. From the first relation of , we have that $$\label{1eq:3.9bis}
\un\b=0 \,\text{ for $N=2,3$,}$$ and, for $N\geq 4$, $$\label{1eq:3.9bisbis}
    \un\b>0 \quad \text{ satisfies } \quad  h_u(\un\b R)=\frac{\mu R}{D}\frac{d(N-3)}{\nu R+ d(N-3)}.$$ Thanks to all these properties and the implicit function theorem, there exists an analytic function $\d(\b):(\un\b,\ov\b)\to(0,+\infty)$ which satisfies $\chi_v(\d(\b))=\chi_u(\b)$. By differentiating this relation we obtain $$\label{1eq:3.9ter}
\chi'_v(\d(\b))\d'(\b)=\chi_u'(\b),$$ therefore $\d(\b)$ is increasing and, thanks to , satisfies $$\label{1eq:3.10}
\lim_{\b\da \un\b}\d(\b)=0, \quad \lim_{\b\ua \ov\b}\d(\b)=+\infty, \quad \lim_{\b\da \un\b}\d'(\b)=\begin{cases}0 & \text{if $N=2,3$} \\
\frac{\chi'_u(\un\b)}{d}>0 & \text{if $N=4$} \\
+\infty & \text{if $N\geq 5$.}  \end{cases}$$

This analysis allows us to reduce the search for solutions (respectively, supersolutions) of to the search for solutions of the following system for the variables $\a,\b$ and the parameter $c$ (respectively, the corresponding one with the $\lq\lq\geq"$ signs) $$\label{1eq:3.11}
\left\{\begin{array}{l}
c\a-D\a^2+D\b^2=g'(0) \\
c\a-d\a^2-d\d^2(\b)=f'(0). \\
\end{array}\right.$$ In the first quadrant of the $(\b,\a)$ plane, the first equation describes an hyperbola $\Si_D(c)$ that can be parameterized through the graphs $$\label{1eq:3.13}
\a_D^{\pm}(c,\b):=\frac{c\pm\sqrt{c^2-c_g^2+4D^2\b^2}}{2D},$$ which are defined for $\b\geq\hat{\b}(c)$, where $$\label{1eq:3.13bis}
\hat{\b}(c):=\min\{\b\geq 0:c^2\geq c_g^2-4D^2\b^2\}.$$ Observe that $c\mapsto\hat{\b}(c)$ is continuous, non-increasing, and that $$\hat{\b}(c)
=\sqrt{\frac{g'(0)}{D}}  \text{ for $c=0$}, \quad \hat{\b}(c)\in\left(\!0,\sqrt{\frac{g'(0)}{D}}\right)  \text{ for $c\in(0,c_g)$}, \quad
\hat{\b}(c)=0 \text{ for $c\geq c_g$}.$$ Note also that $\a_D^{-}\left(c,\sqrt{\frac{g'(0)}{D}}\right)=0$ for every $c>0$.

The second equation of represents, instead, a curve which will be denoted by $\Si_d(c)$ and can be parameterized through the positive functions $$\label{1eq:3.12}
\a_d^\pm(c,\b)=\frac{c\pm\sqrt{c^2-c_f^2-4d^2\d^2(\b)}}{2d}.$$ Thus, $\a_d^\pm(c,\b)$ is not defined for $c< c_f$, for $c=c_f$ it consists of the point $\left(\un\b,\frac{c}{2d}\right)$, and, for $c>c_f$, if we set $\breve{\b}(c)\in(\un\b,\ov\b)$ to be the unique value of $\b$ such that $c^2=c_f^2+4d^2\d^2(\b)$, then $\a_d^{\pm}(c,\b)$ is defined for $\un\b\leq\b\leq\breve{\b}(c)$ and looks like a deformed arc of circle which is symmetric about $\a=\frac{c}{2d}$.

By analyzing the behavior of these curves with respect to $c$, we have that, for $i\in\{d,D\}$ and within their domain of definition in the first quadrant of the $(\b,\a)$ plane, $$\label{1eq:3.14}
c\mapsto \a_i^+(c,\b) \text{ increases to $+\infty$}, \quad             c\mapsto \a_i^-(c,\b) \text{ decreases to $0$}.$$

Finding solutions of corresponds to find intersections between the curves $\Si_D(c)$ and $\Si_d(c)$ respectively, while finding supersolutions amounts to find intersections between the region lying between the curves $\a_D^\pm$, which will be denoted by $\mathcal{S}_D(c)$, and the region lying between $\a_d^\pm$, which will be denoted by $\mathcal{S}_d(c)$.

Observe that a necessary condition for the existence of this kind of (super)solutions is $c$ to be not smaller than $c_f$, in order $\mathcal{S}_d(c)$ to be non-empty. If the curve $\Si_d(c)$ arises, for $c=c_f$, inside $\mathcal{S}_D(c)$, which is equivalent to $$\label{1eq:3.15}
\a_D^-(c_f,\un\b)\leq \frac{c_f}{2d}\leq \a_D^+(c_f,\un\b),$$ then supersolutions of the considered form will exist for all $c\geq c_f$ and, in this case, we set $c^*:=c_f$. Otherwise, we define $c^*$ as the first value of $c> c_f$ for which $\Si_D(c)$ and $\Si_d(c)$ intersect. Observe that such a value of $c$ exists by continuity and thanks to . Moreover, also guarantees that supersolutions to will exist for every $c\geq c^*$.

**<span>Claim.</span> If $N\leq 5$ and $c^*>c_f$, then the curves $\Si_D(c^*)$ and $\Si_d(c^*)$ are tangent.**

Using the definition of $c^*$ and the regularity of the curves, the claim will follow if we show that $\b^*>\un\b$ at any intersection point $(\b^*,\a^*)$ between $\Si_D(c^*)$ and $\Si_d(c^*)$.

Let us assume by contradiction that $\b^*=\un\b$ (we implicitly consider $\a_D^\pm(c^*,\un\b)$ to be defined, otherwise no intersection can occur). By evaluating some of the derivatives of the curves with respect to $\b$ at $(c^*,\un\b)$, we will show that $c^*$ is not the first value of $c$ for which the curves intersect, obtaining a contradiction.

We consider the following derivatives $$\begin{gathered}
    \p_\b\,\a_d^\pm(c,\b)=\mp\frac{2d\d(\b)\d'(\b)}{\left(c^2-c_f^2-4d^2\d^2(\b)\right)^{1/2}}, \label{1eq:3.18a}\\ \p^2_{\b\b}\,\a_d^\pm(c,\b)=\mp2d\frac{\left(\d'(\b)^2+\d(\b)\d''(\b)\right)\left(c^2-c_f^2\right)-4d^2\d^3(\b)\d''(\b)}{\left(c^2-c_f^2-4d^2\d^2(\b)\right)^{3/2}},
     \label{1eq:3.18b}\\
    \p_\b\,\a_D^\pm(c,\b)=\pm\frac{2D\b}{\left(c^2-c_g^2+4D^2\b^2\right)^{1/2}}, \label{1eq:3.19a}\\ \p^2_{\b\b}\,\a_D^\pm(c,\b)=\pm\frac{2D\left(c^2-c_g^2\right)}{\left(c^2-c_g^2+4D^2\b^2\right)^{3/2}}, \label{1eq:3.19b}
 \end{gathered}$$ and assume for example that $\a_d^-(c^*,\un\b)=\a_D^+(c^*,\un\b)$ (the case $\a_d^+(c^*,\un\b)=\a_D^-(c^*,\un\b)$ can be treated analogously).

For $N\in\{2,3\}$, i.e. when $\un\b=0$, , and give that $\p_\b\,\a_d^-(c^*,0)=0=\p_\b\,\a_D^+(c^*,0)$, thus we consider the second derivatives of such curves with respect to $\b$. To this end, we have to compute $\lim_{\b\da0}\d''(\b)$: we invert , obtaining $$\d(\b)=\frac{1}{R}k_v\left(\frac{R}{d}\chi_u(\b)\right),$$ where $k_v:(0,+\infty)\to(0,+\infty)$ is the inverse function of $h_v$ introduced in , and elementary differentiations give $$\label{1eq:3.20}\d''(\b)=\frac{R}{d^2}k_v''\left(\frac{R}{d}\chi_u(\b)\right)\chi_u'^2(\b)+\frac{1}{d}k_v'\left(\frac{R}{d}\chi_u(\b)\right)\chi_u''(\b),$$ with $$\label{1eq:3.22} k_v'(h_v(r))=\frac{1}{h_v'(r)}, \quad \text{and} \quad k_v''(h_v(r))=-\frac{h_v''(r)}{h_v'(r)^3}.$$ On the one hand, , entails $\chi_u'(0)=0$ and $\chi_u''(0)\in(0,+\infty)$; on the other hand, for $N=2$, implies that $k_v(r)=r/d$, while, for $N=3$, and imply $\lim_{r\da0}k_v'(r)=\lim_{r\da0}k_v''(r)=0$. Thus, gives, in both cases, $\lim_{\b\da 0}\d''(\b)\in[0,+\infty)$ and, as a consequence, $\lim_{\b\da 0}\d(\b)\d''(\b)=0$. From and , we obtain $$\p^2_{\b\b}\a_d^-(c^*,0)=0<\frac{2D}{\sqrt{c^{*2}-c_g^2}}=\p^2_{\b\b}\a_D^+(c^*,0),$$ thus, by continuity, $\a_d^-(c^*,\b)$ will lie below $\a_D^+(c^*,\b)$ for $\b\sim 0$. But $\a_d^+(c^*,\b)$ lies, again for $\b\sim 0$, above $\Si_D(c^*)$, and this means that $c^*$ is not the first value of $c$ for which $\Si_d(c)$ and $\Si_D(c)$ intersect, which gives the desired contradiction.

For $N=4$, we have $\un\b>0$ and gives that $\lim_{\b\da\un\b}\d(\b)\d'(\b)=0$. The same holds true for $N=5$ since, from , we have $$\label{1eq:3.24}
\frac{\chi_v'(\d(\b))}{\d(\b)}\d(\b)\d'(\b)=\chi_u'(\b),$$ and, as $\b\da\un\b$, the right-hand side converges to $\chi_u'(\un\b)>0$, while the first factor on the left-hand side converges to $\chi_v''(0)$, which, thanks to , equals $+\infty$. Thus, for $N=4,5$, and give $$\p_{\b}\a_d^-(c^*,\un\b)=0<\frac{2D\un\b}{\sqrt{c^{*2}-c_g^2+4D^2\un\b^2}}=\p_{\b}\a_D^+(c^*,\un\b),$$ and we obtain a contradiction by reasoning on the relative positions of the curves, as before.

To conclude, we observe that, for $N\geq 6$, thanks again to , we have $\chi_v''(0)=\frac{2dR}{N-5}$, thus, using , $\lim_{\b\da\un\b}\d(\b)\d'(\b)\in(0,+\infty)$, and we cannot derive an immediate contradiction as above, valid for general values of the parameters.

As done in Section \[0section1\], the next step is the construction of compactly supported, small, stationary subsolutions of for $c<c^*$, $c\sim c^*$. The first result covers the case in which $c^*=c_f$.

\[1pr:3.1\] For every $0<c<c_{f}$, there exist $\e_0=\e_0(c)>0$ and $L=L(c)>R+1$ such that, for $0<\e<\e_0$, problem admits a compactly supported subsolution of the form $\e(0,\un v)$, where $$\un v:=\left\{\begin{array}{ll}
    e^{\frac{c}{2d}x}\cos(\a x)\psi(y) & \text{in $(-\frac{\pi}{2\a},\frac{\pi}{2\a})\times A_L$} \\
    0 & \text{otherwise,}
    \end{array}\right.$$ $A_L:=\{y\in{{\mathbb{R}}}^{N-1}: R+1<|y|<L\}$, $\psi$ is the positive solution of the following problem $$\left\{\begin{array}{ll}
    -\D_y\psi=\l_1(L)\psi & \text{in $A_L$} \\
    \psi=0 & \text{on $\p A_L$} \\
    \max_{A_L}\psi=1, &
    \end{array}\right.$$ and, once fixed a small enough $\t>0$, $$\a=\sqrt{\frac{1}{d}\left(f'(0)-\frac{c^2}{4d}-\t-d\l_1(L)\right)}.$$

The proof follows as the one of Proposition \[0pr:1.3\], once observed that $\un v$ solves $-d\D v+cv_x=(f'(0)-\t)v$ and $\l_1(L)\to+\infty$ as $L\to+\infty$.

As a second result, we construct arbitrarily small, compactly supported, stationary subsolutions to for $c<c^*$, $c\sim c^*$, when $c^*>c_f$. The key point will be the Claim proved in the construction of the supersolutions; for this reason the limitation $N\leq 5$ arises.

\[1pr:3.2\] Assume $N\leq 5$ and $c^*>c_f$. Then, for $c<c^*$, $c\sim c^*$, there exists arbitrarily small, non-negative, compactly supported, stationary subsolutions of .

As in Proposition \[0pr:1.5\], we look for solutions of a penalized version of the linearizion , i.e. with $g'(0)$ and $f'(0)$ respectively replaced by $g'(0)-\t$ and $f'(0)-\t$, $\t\sim 0$. In addition, we require $v$ to vanish at $|y|=L\gg R$. Thus, in analogy with the previous construction of this section, such solutions will be sought of the form $$(\un u,\un v)= e^{\a x}|y|^{-\tau}\left(\frac{\nu J_{\tau}(\b |y|)}{\mu J_{\tau}(\b R)-D\b J_{\tau+1}(\b R)},\frac{K_{\tau}(\d |y|)I_{\tau}(\d L)-I_{\tau}(\d |y|)K_{\tau}(\d L)}{K_{\tau}(\d R)I_{\tau}(\d L)-I_{\tau}(\d R)K_{\tau}(\d L)}\right),$$ where $I(r)$ is the modified Bessel function of first kind (see Appendix \[sectionapp1\]). Observe that $\un v$ is positive for $|y|<L$, since the function $r\mapsto r^{-\tau}(K_{\tau}(\d r)I_{\tau}(\d L)-I_{\tau}(\d r)K_{\tau}(\d L))$ is decreasing thanks to and , and vanishes for $r=L$.

With similar computations as in the first part of this section, the search for such solutions can be reduced to the search for intersections between the curve $\tilde\Si_D(c)$, which is obtained by replacing $g'(0)$ in $\Si_D(c)$ by $g'(0)-\t$, and the curve $$\tilde{\Si}_d(c,L):=\{(\b,\tilde\a_d^\pm(c,\b,L))\},$$ where $$\tilde\a_d^\pm(c,\b,L):=\frac{c\pm\sqrt{c^2-c_f^2-4d^2\tilde{\d}^2(\b,L)}}{2d},$$ and $\tilde{\d}(\b,L)$ is implicitly defined through $$d\tilde\d\frac{K_{\tau+1}(\tilde\d R)I_{\tau}(\tilde\d L)+I_{\tau+1}(\tilde\d R)K_{\tau}(\tilde\d L)}{K_{\tau}(\tilde\d R)I_{\tau}(\tilde\d L)-I_{\tau}(\tilde\d R)K_{\tau}(\tilde\d L)}=\frac{\nu D\b
        J_{\tau+1}(\b R)}{\mu J_{\tau}(\b R)-D\b J_{\tau+1}(\b R)}.$$ Since $\frac{K_{\tau}(r)}{I_{\tau}(r)}\to 0$ as $r\to+\infty$, thanks to and , it is easy to see that, as $L\to +\infty$, $\tilde{\d}(\cdot,L)$ converges, locally uniformly for $\b\in(\un\b,\ov\b)$ to the $\d(\cdot)$ implicitly defined by .

As shown in the Claim above, the tangency point between $\Si_D(c^*)$ and $\Si_d(c^*)$ satisfies $\b^*>\un\b$, thus, by continuity, the curves $\tilde{\Si}_d(c,L)$ and $\tilde{\Si}_D(c)$ have no real intersections in a neighborhood of $(\b^*,\a^*)$, for sufficiently large $L$, small $\t$, and $c<c^*$, $c\sim c^*$. Using Rouché's theorem as in the proof of Proposition \[0pr:1.5\], it is possible to find complex intersections for $c<c^*$, $c\sim c^*$ and, consequently, to construct the desired subsolutions, again along the same lines of the proof of Proposition \[0pr:1.5\].

With these elements, the proof of parts and of Theorem \[1th:3.3\], which assert that $c^*$ is the asymptotic speed of propagation of in the $x$ direction, follows in the same lines as the one of Theorem \[0th:1.4\].

Properties of the asymptotic speed of propagation of  {#1section4}
-----------------------------------------------------

The main goal of this section is to prove, among other results, the rest of Theorem \[1th:3.3\] and Theorem \[1th:5.1\], i.e. we will deal with the study the qualitative behavior of the speed of propagation $c^*$ as several parameters appearing in vary. For this reason, we will explicitly point out the dependence of the quantities introduced in the previous section on the parameters taken into consideration in each case. Moreover, unless differently specified, we will consider $2\leq N\leq 5$, i.e. the values of the dimensions for which we have been able to establish a precise geometrical characterization of $c^*$.

By construction, we know that $c^*\geq c_f$. The first result of this section establishes for which values of the parameters $c^*$ coincides with $c^*_f$, i.e. when the cylinder has no effect on the propagation. As an immediate consequence, we will obtain the complementary result on then $c^*$ is strictly larger than $c_f$, i.e. when the presence of diffusion and reaction heterogeneities inside the cylinder enhance the global propagation with respect to the homogeneous case.

\[1pr:4.1\] The asymptotic speed of propagation of problem satisfies $c^*>c_f$ if and only if $$\label{1eq:4.1new}
    D(f'(0)-d\un\b^2(D))> d(2f'(0)-g'(0)),$$ where $\un\b(D)$ is the one given in –. More precisely:

(i) \[1pr:4.1i\] if $N=2,3$, $c^*(D)>c_f$ if and only if $$\label{1eq:4.1bis}
            \frac{D}{d}>2-\frac{g'(0)}{f'(0)};$$

(ii) \[1pr:4.1ii\] if $N=4,5$ and $R\sqrt{\frac{f'(0)}{d}}\geq j_\tau$, then there exists $\ov D\geq 0$ such that $c^*(D)>c_f$ if and only if $D>\ov D$. In particular, if $2f'(0)\leq g'(0)$, then $\ov D=0$, otherwise $\ov D>0$;

(iii) \[1pr:4.1iii\] if $N=4,5$ and $R\sqrt{\frac{f'(0)}{d}}<j_\tau$, then, by setting $M:=\min_{D>0}D(f'(0)-d\un\b^2(D))$, we have $M\in(-\infty,0)$ and

    1.  if $2f'(0)-g'(0)<\frac{M}{d}$ or $2f'(0)-g'(0)\geq 0$ , then there exists $\ov D\geq 0$ such that $c^*(D)>c_f$ if and only if $D>\ov D$. In the former case, $\ov D=0$, while in the latter $\ov D>0$;

    2.  if $\frac{M}{d}\leq 2f'(0)-g'(0)<0$, then there exist $0<D_1\leq D_2<+\infty$ such that $c^*(D)=c_f$ if and only if $D\in[D_1,D_2]$.

As already pointed out in Section , $c^*=c_f$ if and only if holds true, which can be easily shown to be equivalent to the complementary condition to . Part then follows directly by recalling from that $\un\b(D)\equiv 0$ if $N=2,3$. When $N=4,5$, instead, we have to study the function $\un\b(D)>0$. From its definition in , we have $$\label{1eq:4.4}
            \un\b(D)=\frac{1}{R}k_u\left(\frac{\mu R}{D}\frac{d(N-3)}{\nu R+ d(N-3)}\right),$$ where $k_u:(0,+\infty)\to(0,j_\tau)$ is the inverse of the function $h_u$ introduced in . Thus, $\un\b(D)$ is continuous, positive, decreasing, and satisfies $$\label{1eq:4.4bis}
            \lim_{D\da0}\un\b(D)=\frac{j_\tau}{R}, \qquad \lim_{D\to+\infty}\un\b(D)=0.$$ As a consequence, the function $\zeta(D):=D\left(f'(0)-d\un\b^2(D)\right)$ converges to $0$ as $D\da 0$ and to $+\infty$ as $D\to+\infty$. In addition, we now prove that $\zeta$ is convex. Indeed, from , we have $$\begin{aligned}
\zeta''(D)&=-2d\left(2\un\b(D)\un\b'(D)+D\un\b'^{2}(D)+D\un\b(D)\un\b''(D)\right) \notag \\
&=-2d\frac{\mu^2A^2}{D^3}\left(k_u'^{2}\left(\frac{\mu R A}{D}\right)+k_u\left(\frac{\mu R A}{D}\right)k_u^{''}\left(\frac{\mu R A}{D}\right)\right) 
        \end{aligned}$$ where we have set $A:=\frac{d(N-3)}{\nu R+d(N-3)}\in(0,1)$. Thus, thanks to the analogous relation of satisfied by $k_u$, showing that $\zeta''(D)>0$ for $D>0$ amounts to prove that $\frac{rh_u''(r)-h_u'(r)}{h_u'^3(r)}>0$ for $r\in~(0,j_\tau)$, which holds true, since both the numerator and the denominator are positive, the former by Proposition \[pr:A.3\] and the latter by Lemma \[le:A.1\].

As a consequence, under the assumption of part , $\zeta'(0)=f'(0)-d\frac{j_\tau^2}{R^2}\geq 0$, and the existence and properties of $\ov D$ now follow immediately.

On the contrary, with the assumption of part , $\zeta'(0)<0$, and there exists $D_0>0$ such that $\zeta(D)$, which still vanishes for $D=0$, is negative for $D<D_0$, positive for $D>D_0$ and has a unique critical point which is a global minimum. With this analysis, the other properties readily follow.

\[1re:4.2\]

(i) \[1re:4.20\] From the previous proposition, and , we immediately obtain the proof of Theorem \[1th:3.3\] by setting $C:=\frac{D\un\b^2}{f'(0)}.$

(ii) \[1re:4.2i\] Observe that condition for $c^*$ to be greater than $c_f$ is the same found for $c^*_\infty$ (which in turn is the same found for the planar road-field problem , as already remarked).

(iii) \[1re:4.2ii\] Case (b), which occurs, for example, for $R\sim 0$ and $2f'(0)<g'(0)\leq2f'(0)-\frac{M}{d}$ (observe that $M$ does not depend on $g'(0)$), shows that the set of $D$'s for which the speed of propagation is enhanced can be non-connected. This is a remarkable difference with respect to both the other cases described by Proposition \[1pr:4.1\] and $c^*_\infty$ (compare, e.g., with Figure \[0fig:1.1\]).

(iv) \[1re:4.2iii\] Another consequence of case (b) in Proposition \[1pr:4.1\] is that $c^*(D)$ can be decreasing in certain ranges of $D$, in contrast with $c^*_\infty$ and ${c_{\mathrm{rf}}}$, which are non-decreasing in $D$, as proved in Proposition \[0pr:1.6\] and as it can be seen from the construction of ${c_{\mathrm{rf}}}$ given in @BRR2.

(v) \[1re:4.2iv\] By combining the construction of supersolutions of Section and Proposition \[1pr:3.1\], we can give a characterization of $c^*$ also for $N\geq 6$; precisely we obtain that, if the complementary condition of holds true, which reads $$\label{1eq:4.1old}
                D(f'(0)-d\un\b^2(D))\leq d(2f'(0)-g'(0)),$$ then $c^*=c_f$ is the asymptotic speed of propagation of problem .

Despite the fact that we are not able to characterize precisely $c^*$ for $N\geq 6$, Remark \[1re:4.2\] gives us a sufficient condition, with no restriction on the spatial dimension, for $\O$ not to have any effect on the propagation. In the following result, we use such a result to show that, fixing all the other parameters, $c^*(N)=c_f$ for $N$ sufficiently large. This can be interpreted by considering any cylinder with section $B_{N-1}(r)$, with $r>R$ so that it contains $\O$, and observing that the ratio of the volumes of the sections is given by $\left(\frac{r}{R}\right)^{N-1}$, entailing that the volume that has to be filled by the solution of grows to $+\infty$ as $N\to+\infty$, or, the other way around, the relative volume of $\O$ becomes negligible.

\[1pr:4.2bis\] The other parameter being fixed, there exists $N_0\geq 4$ such that, for every $N\geq N_0$, $c^*(N)=c_f$.

Observe that the argument of $k_u$ in converges to $\mu R/D$ as $N\to+\infty$. In addition, implies that $h_u$ converges to $0$ locally uniformly as $N\to+\infty$, thus its inverse $k_u$ converges to $+\infty$ locally uniformly. This, together with , shows that $$\label{1eq:4.4quater}
        \un\b(N)\to+\infty \quad \text { as $N\to+\infty$,}$$ and Remark \[1re:4.2\] allows us to conclude.

We pass now to the study of the asymptotic behavior of $c^*$ as $D$ varies, proving parts and of Theorem \[1th:3.3\].

\[Proof of Theorem \[1th:3.3\] and \] (iv) Observe that, by Proposition \[1pr:4.1\], $c^*(D)>c_f$. for large $D$. First of all, we prove that $c^*(D)$ is unbounded. Assume by contradiction that $c^*(D)$ is bounded for $D\to+\infty$. Then, for large $D$, $c^*(D)<c_g=2\sqrt{Dg'(0)}$ and $\frac{c^*(D)}{2D}<\frac{c^*(D)}{2d}$. These considerations entail that, for $c=c^*(D)$, the curves $\a_d^-(c,\b)$ and $\a_D^+(c,\b)$ are tangent at a point satisfying $\b>0$.

In order to get a contradiction we perform the change of variable $\b'=\b\sqrt{D}$. In this way, we have that the curve $$\a_D^+\left(c^*(D),\frac{\b'}{\sqrt{D}}\right)=\frac{c^*(D)+\sqrt{c^{*2}(D)-4D(g'(0)-\b'^2)}}{2D}$$ is defined for $\b'$ greater than or equal to a value $\hat\b'(c^*(D))$ converging to $\sqrt{g'(0)}$ as $D\to+\infty$, and it goes to $0$ locally uniformly in $\b'$ as $D\to+\infty$. Regarding the other curve, we have to study the behavior of $\d\left(\frac{\b'}{\sqrt{D}},D\right)$, which, according to , is given through $$\chi_v(\d)=\frac{\nu D h_u\left(\frac{\b'}{\sqrt{D}} R\right)}{\mu R - D h_u\left(\frac{\b'}{\sqrt{D}} R\right)}.$$ Thanks to , the right-hand side converges, as $D\to+\infty$, to $\frac{\nu\b'^2R}{\mu(N-1)-\b'^2R}$ locally uniformly in $\b'$. Thus the curve $\a_d^-\left(c^*(D),\frac{\b'}{\sqrt{D}}\right)$ converges for $D\to+\infty$ to a curve that lies away from $\{\a=0\}$, and cannot be tangent to $\a_D^+\left(c^*(D),\frac{\b'}{\sqrt{D}}\right)$, giving the desired contradiction.

Next, we establish the order of convergence of $c^*(D)$ to $+\infty$. By performing the same change of variable $\b'=\b\sqrt{D}$ as above, we observe that the curves $\a_D^\pm\left(c,\frac{\b'}{\sqrt{D}}\right)$ converge to $0$ as $D\to+\infty$, while $\a_d^\pm\left(c,\frac{\b'}{\sqrt{D}}\right)$ converges to a curve lying away from $\{\a=0\}$, thus, for large $D$, the tangency at $c=c^*(D)$ occurs between $\a_D^+$ and $\a_d^-$. Such a tangency condition can be rewritten as $$\begin{gathered}
            \frac{1}{D}\left(1+\sqrt{1-\frac{4Dg'(0)}{c^{*2}(D)}+\frac{4D\b^{'2}}{c^{*2}(D)}}\right)=\frac{1}{d}\left(1-\sqrt{1-\frac{c_f^2}{c^{*2}(D)}-\frac{4d^2\d^{2}\left(\frac{\b'}{\sqrt{D}},D\right)}{c^{*2}(D)}}\right) \iff \\
            \frac{c^{*2}(D)}{D}\left(1+\sqrt{1-\frac{4Dg'(0)}{c^{*2}(D)}+\frac{4D\b^{'2}}{c^{*2}(D)}}\right)=\frac{4f'(0)+4d\d^{2}\left(\frac{\b'}{\sqrt{D}},D\right)}{1+\sqrt{1-\frac{c_f^2}{c^{*2}(D)}-\frac{4d^2\d^{2}\left(\frac{\b'}{\sqrt{D}},D\right)}{c^{*2}(D)}}}
        \end{gathered}$$ and, since the last term converges to a positive constant as $D\to+\infty$, this implies that the same occurs for $c^{*2}(D)/D$, as we wanted.

\(v) The last claim is trivial, since, when holds true for $D\sim 0$, then Proposition \[1pr:4.1\] ensures that $c^*(D)\equiv c_f$ for $D\sim 0$.

We pass now to the opposite case in which holds true for $D\sim 0$. We know that $\Si_d(c)$ arises, for $c=c_f$, outside ${\mathcal}{S}_D(c)$ and, since $\a_D^+(c,\b)$ converges to $+\infty$ as $D\da 0$, this implies that $\Si_d(c)$ arises, in the first quadrant, below $\a_D^-(c,\b)$. As a consequence, for $c=c^*(D)$, $\a_d^+(c,\b)$ is tangent to $\a_D^-(c,\b)$. Regarding the latter curve, we have $$\label{1eq:4.4-5}
            \a_D^-(c,\b)=\frac{2(g'(0)-D\b^2)}{c+\sqrt{c^2-c_g^2+4D^2\b^2}}\to\frac{g'(0)}{c}$$ as $D\da 0$, locally uniformly in $\{\b\geq 0\}$.

In order to conclude, we study the behavior of $\a_d^+(c,\b,D)$ as $D\to 0$. To this end, we observe that $$\label{1eq:4.5bis}
            \ov\b(D)\to \frac{j_\tau}{R} \text{ for } D\to 0,$$ where $\ov\b$ is the one defined in .

When $N=2,3$, since $\chi_u(\b)$ and $h'_u(r)$ converge to $0$, for $D\to 0$, locally uniformly in $(0,j_\tau/R)$, we have that $\d(\b)$ and $\d'(\b)$ converge to $0$, and $\a_d^+(c,\b,D)$ converges to the horizontal segment $\frac{c+\sqrt{c^2-c_f^2}}{2d}$ locally uniformly in $(0,j_\tau/R)$.

For $N=4,5$, and give that $\a_d^+(c,\b,D)$ converges, for $D\to 0$, to the vertical segment $\left\{\left(\frac{j_\tau}{R},\a\right):\frac{c}{2d}\leq\a\leq\frac{c+\sqrt{c^2-c_f^2}}{2d}\right\}$.

This, together with the geometric characterization of $c^*(D)$ and , entails that $c_0$ is the unique value of $c$ for which $$\frac{c+\sqrt{c^2-c_f^2}}{2d}=\frac{g'(0)}{c},$$ which is given by the expression in (we remark that such a value is well defined, since Proposition \[1pr:4.1\] implies that, when $c^*(D)>c_f$ for $D\sim 0$, necessarily we have $g'(0)\geq 2f'(0)>f'(0)$). The fact that $c_0=c_f$ if $g'(0)=2f'(0)$ (resp. $c_0>c_f$ if $g'(0)>2f'(0)$) can be obtained though simple computations, using the expression of these quantities.

By comparing with the diagram in Figure \[0fig:1.1\] and recalling the expression of $c_a$ given in , we readily observe that the limit of $c^*(D)$ as $D\to 0$ coincides with the same limit for $c^*_\infty$. Indeed, using , we observe that condition for the enhancement reduces to $g'(0)/f'(0)> 2$ for $D=0$, the same for $c^*_\infty$.

We pass now to study the qualitative properties of the speed of propagation $c^*$ as a function of the radius of the cylinder $\O$, giving the proof of Theorem \[1th:3.3\].

\[Proof of Theorem \[1th:3.3\]\] Recall that, when $c^*>c_f$, $c^*$ is the first value of $c$ for which the curves $\Si_D(c)$ and $\Si_d(c)$ are tangent. On the one hand, $\Si_D(c)$ does not depend on $R$; on the other one, we will now show that $$\label{1eq:monotoniaR}
        R\mapsto \a_d^+(c,\b,R) \text{ is decreasing}, \quad R\mapsto \a_d^-(c,\b,R) \text{ is increasing},$$ which will imply the desired monotonicity. In order to prove , we show that, in its domain of definition, $R\mapsto\d(\b,R)$ is increasing. To this end, observe that , which implicitly defines $\d(\b,R)$, can be rewritten as $$d\d \frac{K_{\tau+1}(\d R)}{K_{\tau}(\d R)}=\frac{\nu D\b}{\mu \frac{J_{\tau}(\b R)}{J_{\tau+1}(\b R)}- D \b }.$$ The right-hand side increases with $R$ thanks to Lemma \[le:A.1\], while implies that the left-hand side is non-increasing in $R$. This, together with the fact that $\chi_v$ is increasing in $\d$, allows to conclude .

*Proof of the first limit in .* For $N=2,3$, we have that the threshold for enhancement, given by , does not depend on $R$. Thus, if does not hold true, then $c^*(R)\equiv c_f$ for every $R>0$ and the first limit in trivially follows.

Otherwise, from , we observe that $$\label{1eq:4.10bis}
        \ov\b(R)=\frac{1}{R}k_u\left(\frac{\mu R}{D}\right);$$ thus, implies that $\ov\b(R)\to+\infty$ for $R\da 0$. Moreover, from we obtain $$\label{1eq:4.11}
        \d(\b,R)=\frac{\chi_u(\b,R)}{d}\frac{k_v\left(\frac{R\chi_u(\b,R)}{d}\right)}{\frac{R\chi_u(\b,R)}{d}},$$ where $k_v:(0,+\infty)\to(0,+\infty)$ is the inverse function of $h_v$, and we observe from that $$\label{1eq:4.12}
        \lim_{R\da 0}\chi_u(\b,R)=0 \quad \text{locally uniformly in $\b$.}$$ Thus, the first factor of the right-hand side of tends to 0, while the second one to $k_v'(0)$, which, from and , can be shown to be a positive constant for $N=2$ and equal to $0$ for $N=3$.

We conclude that $\lim_{R\da 0}\d(\b,R)$=0 locally uniformly in $\b$ and, therefore, the graphs of the curves $\a_d^{\pm}$ converge, in the same way, to the horizontal lines $\frac{c\pm\sqrt{c^2-c_f^2}}{2d}$. Thus, for every $c>c_f$, there are intersection with $\Si_D(c)$ for $R\sim 0$, and, as a consequence, $c_f<c^*<c$, which concludes the proof in this case.

For $N=4,5$, we recall that $\un\b(R)$ is defined as the unique positive solution smaller than $j_\tau/R$ of $$R\chi_u(\un\b(R),R)=d(N-3).$$ If $\un\b(R)$ was bounded as $R\da 0$, such a relation, together with , would give a contradiction. Thus $\lim_{R\da 0}\un\b(R)=+\infty$, which entails that holds true for $R\sim 0$. As a consequence, $c^*(R)\equiv c_f$ for small $R$, and the limit trivially follows.

*Proof of the second limit in .* First of all observe that, if the first condition in the definition of $c^*_\infty$ in holds true, then is also satisfied. Thus, in this case, $c^*(R)=c_f=c^*_\infty$ for every $R$, and the same obviously holds true for the limit $R\to+\infty$.

In addition, since $\d(\b,R)$, is defined for $\b\in(\un\b(R),\ov\b(R))$ and $\ov\b(R)<j_\tau/R$, we have that the domain of $\d(\b,R)$ shrinks to $0$ as $R\to+\infty$, and the curve $\Si_d(c,R)$ collapses to the vertical segment $$\label{1eq:4.12bis}
        \left\{(0,\a):\frac{c-\sqrt{c^2-c_f^2}}{2d}\leq \a\leq\frac{c+\sqrt{c^2-c_f^2}}{2d} \right\}.$$ Finally, we observe that the extrema of such a segment are exactly the quantities $r_d^\pm(c)$ introduced in and used in the construction of $c^*_\infty$, while the endpoints of the region ${\mathcal}S_D(c)\cap\{\a=0\}$ coincide with $r_D^\pm$ (see again ). This shows that $\lim_{R\to+\infty}c^*(R)$ can be characterized by repeating exactly the construction of $c^*_\infty$ performed in Proposition \[0pr:1.2bis\] and, therefore, that the two quantities coincide in all the cases.

In the last part of this section, we consider problem in the specific case $N=2$. Up to a translation in the $y$ direction and using the symmetry of the domain and of the equations, it reduces to $$\label{1eq:5.1}
\left\{
\begin{array}{ll}
u_t-D\D u=g(u) & \text{in ${{\mathbb{R}}}\times(-R,0)\times{{\mathbb{R}}}^+$} \\
v_t-d\D v=f(v) & \text{in ${{\mathbb{R}}}\times\{y>0\}\times{{\mathbb{R}}}^+$} \\
D\, u_y = \nu v-\mu u & \text{on ${{\mathbb{R}}}\times\{y=0\}\times{{\mathbb{R}}}^+$}\\
-d\, v_y = \mu u-\nu v & \text{on ${{\mathbb{R}}}\times\{y=0\}\times{{\mathbb{R}}}^+$} \\
u_y=0 & \text{on ${{\mathbb{R}}}\times\{y=-R\}\times{{\mathbb{R}}}^+$}.
\end{array}\right.$$ Our main purpose is to prove Theorem \[1th:5.1\], which establish the relationship between and the road-field problem , through a singular rescaling of the exchange parameter $\mu$ as $R\to 0$.

\[Proof of Theorem \[1th:5.1\].\] Observe that problem is obtained by replacing $\mu$ by $\tilde\mu(R)$ in . By reasoning as in Section for $N=2$ (see – and ), we therefore have that $\tilde c^*(R)$ can be geometrically characterized as the smallest value of $c$ for which the regions delimited by the curves $$\label{1eq:5.4}
        \left\{\begin{array}{l}
            c\a-d\a^2-d\d^2=f'(0), \\
            c\a-D\a^2+D\b^2=g'(0), 
        \end{array}\right.$$ where $\b$ and $\d$ are related through $$\label{1eq:5.4bis}
        d\d=\displaystyle{\frac{\nu D h_u(\b R)}{\tilde\mu( R)R-Dh_u(\b R)}},$$ intersect.

If $\frac{D}{d}\leq 2-\frac{g'(0)}{f'(0)}$ (observe that this threshold does not depend neither on $\mu$, nor on $\tilde\mu(R)$, nor on $R$), we have, from Proposition \[1pr:4.1\] and , that both $\tilde c^*(R)$ and ${c_{\mathrm{rf}}}$ coincide with $c_f$, thus the result trivially holds true.

On the other hand, for $\frac{D}{d}> 2-\frac{g'(0)}{f'(0)}$, we briefly recall that in @BRR2 [Section 4], following the same strategy of this work, ${c_{\mathrm{rf}}}$ is characterized as the smallest $c$ for which the system $$\label{1eq:5.3}
        \left\{\begin{array}{l}
            c\a-d\a^2-d\d^2=f'(0) \\
            c\a-D\a^2=g'(0)-\frac{\mu d \d}{\nu+d\d},
        \end{array}\right.$$ possesses real solutions.

By comparing with , in order to conclude, it is sufficient to prove that, if we use to express $\b$ as a function of $\d$ (and of $R$), we have, locally uniformly in $\{\d>0\}$, $$\label{1eq:5.5}
        D\b^2(\d,R)\to\frac{\mu d \d}{\nu+d\d} \quad \text{ as $R\da 0$.}$$ To this end, we explicitly compute $\b$ from getting $$\b(\d,R)=\frac{1}{R}k_u\left(\frac{\tilde\mu(R)R}{D}\frac{d\d}{\nu+d\d}\right),$$ and, by taking the limit as $R\to 0$, using and , we finally obtain .

The last result of this section complements the ones of Theorem \[1th:3.3\] and Theorem \[1th:5.1\], since we determine the behavior of the speed of propagation $\tilde c^*(R)$ in the $x$ direction of problem for all the possible behaviors of the positive function $\tilde{\mu}(R)$.

\[1pr:5.2\] The asymptotic speed of propagation $\tilde c^*(R)$ in the $x$ direction of problem satisfies:

(i) \[1pr:5.2i\] if does not hold true, $\lim_{R\da 0}\tilde c^*(R)=c_f$;

(ii) \[1pr:5.2ii\] if holds true, then

    -   if $\lim_{R\da0}\frac{\tilde{\mu}(R)}{R}=+\infty$ and $\lim_{R\da 0}\tilde{\mu}(R)R$ exists, then $\lim_{R\da 0}\tilde c^*(R)=c_f$;

    -   \[1pr:5.2iii\] if $\lim_{R\da0}\frac{\tilde{\mu}(R)}{R}=0$, then $\lim_{R\da 0}\tilde c^*(R)=c^*_\infty>c_f$.

\(i) As already remarked, since the threshold given in does not depend on $\tilde\mu$ nor on $R$, we have $\tilde c^*(R)\equiv c_f$.

\(ii) Under the assumption of this part, the curve $\Si_d$ arises, for $c=c_f$, outside $\mathcal{S}_D(c)$. When $\lim_{R\da0}\frac{\tilde{\mu}(R)}{R}=+\infty$ we can reason as in the proof of Theorem \[1th:3.3\] to show that $\ov\b(R)\to+\infty$. Recall, indeed, . By setting $l:=\lim_{R\da 0}\tilde{\mu}(R)R\in[0,+\infty]$, if $l>0$, then, since $k_u\left(\frac{\tilde{\mu}(R)R}{D}\right)$ converges as well to a positive constant as $R\da 0$, the conclusion follows directly from . Otherwise, if $l=0$, gives $\ov\b^2(R)\sim\frac{1}{D}\frac{\tilde{\mu}(R)}{R}\to+\infty$ as $R\da 0$.

Moreover, since $\lim_{R\da0}\frac{\tilde{\mu}(R)}{R}=+\infty$, and imply that $\d(\b,R)\to 0$ locally uniformly for $\{\b\geq 0\}$. Thus, $\Si_d$ converges locally uniformly to two horizontal lines, and it intersects $\mathcal{S}_D(c)$ for every $c>c_f$, which gives the desired result.

Finally, when $\lim_{R\da0}\frac{\tilde{\mu}(R)}{R}=0$, by using with $\mu$ replaced by $\tilde{\mu}(R)$ and , we have that $\ov\b(R)\to 0$ as $R\da 0$, thus $\Si_d(c)$ converges to the vertical segment . Thanks to the assumption of part , such a segment arises, for $c=c_f$, outside $\mathcal{S}_D(c)$, and we can reason as in the proof of Theorem \[1th:3.3\] to show that $\lim_{R\da0}\tilde c^*(R)=c^*_\infty$.

The case of mortality for $v$ {#section4}
=============================

In this last section we consider the system with the mortality term and the corresponding one set in two adjacent half-spaces, starting, as done in the first part of the paper, with the latter case, since it is easier.

Since many constructions and arguments are similar to those of the previous sections, we will only detail the main differences.

Two half-spaces with mortality {#section4.1}
------------------------------

When $f(s)=-\r s$, system reads $$\label{0eq:2.1}
\left\{
\begin{array}{ll}
u_t-D\D u=g(u) & \text{in $\{x_N<0\}\times{{\mathbb{R}}}^+$} \\
v_t-d\D v=-\r v & \text{in $\{x_N>0\}\times{{\mathbb{R}}}^+$} \\
D\, \p_{x_N} u = \nu v-\mu u & \text{on $\{x_N=0\}\times{{\mathbb{R}}}^+$}\\
-d\, \p_{x_N} v = \mu u-\nu v & \text{on $\{x_N=0\}\times{{\mathbb{R}}}^+$.}
\end{array}\right.$$ and the long-time behavior for follows the same patterns as system , as shown in the following result.

\[0th:2.1\] Assume . Then, problem admits a unique bounded positive steady state $(U,V)$ which depends only on $x_N$, and for every continuous, bounded and non-negative $(u_0,v_0)$, the solution of starting from $(u_0,v_0)$ satisfies $$\lim_{t\to+\infty}(u,v)=(U,V)$$ locally uniformly in each domain.

The proof follows most of the lines developed in Section \[0section1\]. Indeed, the same super- and subsolutions allow to prove a result like the one given in Proposition \[0pr:1.1\], while for the uniqueness result of the bounded positive steady states with the desired symmetries, it is easy to see that, in this case, the unique possibility for $V$ is $V(x_{N})=V(0)e^{-\sqrt{\r/d}x_{N}}$. Thus, reasoning as in the proof of Proposition \[0pr:1.2\], we obtain that $U$ is decreasing and lies in $(0,1)$ for all $x_N<0$. With these elements, we have that the second relation in has to be replaced by $$V'(0)^2=\frac{\r}{d}V(0)^2,$$ and, by studying as in Proposition \[0pr:1.2\] the monotonicities of $V'(0)^2$ and $V(0)^2$ when $U(0)$ varies in $(0,1)$, we conclude that such a relation has at most one solution.

Turning to the asymptotic speed of propagation, we have the following analogue of Theorem \[0th:1.4\].

\[0th:2.4\] Assume , and recall the quantity $c^*_{m,\infty}$ introduced in . Then, for any $i\in\{1,\ldots,N-1\}$ and every non-negative, continuous, bounded, compactly supported initial datum $(u_0,v_0)$, the solution $(u,v)$ of satisfies:

(i) \[0th:2.4.i\] for all $c>c^*_{m,\infty}$, $\lim_{t\to\infty}\sup_{|x_i|\geq ct}(u,v)=(0,0)$, uniformly in $x_j$, for all $j\neq i$,

(ii) \[0th:2.4.ii\] for all $c<c^*_{m,\infty}$, $\lim_{t\to\infty}\sup_{|x_i|\leq ct}|(u,v)-(U,V)|=0$, locally uniformly in $x_j$, for all $j\neq i$, where $(U,V)$ is the unique steady state of problem given by Theorem \[0th:2.1\].

The proof relies on the construction of super- and subsolutions moving with appropriate speeds, along much the same lines of Section \[0section1\].

Starting with the former ones, it is possible to show that supersolutions of type exist for $c\geq c^*_{m,\infty}$. Indeed, one has to study when the parabolas $$\Si_D(c):=\{D\a^2-c\a +g'(0)=0\}, \qquad \Si_d(c):=\{d\a^2-c\a -\r=0\},$$ have real roots and the regions on the positive $\a$ axis lying between such roots intersect. The parabola $\Si_d(c)$ now has real roots of opposite sign for every $c>0$, and the positive one is given by $r_d^+(c):=\frac{c+\sqrt{c^2+4d\r}}{2d}$. Thus we only have to consider $c=c_g$ and distinguish the case in which $0<\frac{c_g}{2D}\leq r_d^+(c_g)$, which leads to the first case in the definition of $c^*_{m,\infty}$, or $\frac{c_g}{2D}> r_d^+(c_g)$, which leads to the second one.

The construction of compactly supported subsolutions for $0<c<c_g$ follows exactly as in Proposition \[0pr:1.3\], whose proof works also in the case considered here, while, when $\frac{d}{D}>2+\frac{\r}{g'(0)}$, the proof of Proposition \[0pr:1.5\] can be adapted with small modifications to this case, the main difference being that $f'(0)$ has to be replaced by $-\r$ everywhere. This entails that the analogue of the quantity $\un c$ defined in is now equal to $0$ and $\breve\b(c)>0$ for all $c>0$. As a consequence, the curve $\Si_d(c,L)$ converges, as $L\to+\infty$, to $$\Si_d(c,+\infty):=\left\{\left(\b,r_d^+(c)\right):\b<0\right\}\bigcup\left\{(0,\a):0\leq\a\leq r_d^+(c)\right\},$$ which allows us to repeat the same arguments as in Proposition \[0pr:1.5\].

By reasoning as in Section \[0section1\], it is possible to show that the analogue of the qualitative properties of Proposition \[0pr:1.6\]– hold true for $c^*_{m,\infty}$, seen as a function of $d$. These proofs follow, *mutatis mutandis*, as those of Proposition \[0pr:1.6\], thus we do not give the details.

Cylinder with exterior mortality {#2section4}
--------------------------------

We conclude by studying problem ; thus, as in Section \[section3\] we use the notation $x:=x_1$ and $y:=(x_2,\ldots,x_N)$. As we did in the previous sections, we start by analyzing the long-time behavior. The first result provides the counterpart of the upper bound given in Propositions \[0pr:1.1\] and \[1pr:2.1\], and its proof follows exactly along the same lines.

\[2pr:4.1\] Assume , and let $(u,v)$ be the solution of starting with a non-negative, bounded initial datum $(u_0,v_0)$. Then, there exists a non-negative, bounded steady state $(U_2,V_2)$ which is $x$-independent, invariant by rotations about the axis $\{y=0\}$, and such that $$\limsup_{t\to+\infty}(u,v)\leq(U_2,V_2)$$ locally uniformly in each domain.

Regarding the lower bound, things go different here with respect to the previous sections. Indeed, it may happen that the unique steady state of with the symmetries specified in Proposition \[2pr:4.1\] is $(0,0)$. In Propositions \[2pr:4.2\] and \[2pr:4.4\] we will provide a necessary and sufficient condition for the existence of *positive* steady states with the considered symmetries, which will allow us to obtain also a lower bound for the solutions of the parabolic problem .

Before that, we begin with a preliminary discussion, that will lead us to an equivalent problem: the $x$-independent, axially symmetric and bounded stationary solutions of are of the form $(\Phi(r),\Psi(r))$, with such profile functions satisfying $$\label{2eq:stationaryD}
\left\{
\begin{array}{ll}
-D \Phi''-D\frac{N-2}{r}\Phi'=g(\Phi) & r\in(0,R) \\
-d \Psi''-d\frac{N-2}{r}\Psi'=-\r\Psi & r>R \\
D\, \Phi'(R) = \nu \Psi(R)-\mu \Phi(R) & \\
-d\, \Psi'(R) = \mu \Phi(R)-\nu \Psi(R) & \\
\Phi'(0)=0.
\end{array}\right.$$ Since the positive bounded solutions of the second equation in have the form $$\label{2eq:4.1bis}
\Psi(r)=\g r^{-\tau} K_{\tau}\left(\sqrt{\frac{\r}{d}}\,r\right),$$ with $\g>0$ and $\tau=\frac{N-3}{2}$ (see ), the exchange conditions (third and fourth relation in ), together with , then give $$\label{2eq:4.2}
\left\{
\begin{array}{l}
D\Phi'(R)=d\Psi'(R)=-\g R^{-\tau}\sqrt{d\r}K_{\tau+1}\left(\sqrt{\frac{\r}{d}}\,R\right) \\
\mu \Phi(R)=\nu \Psi(R)-d \Psi'(R)=\g R^{-\tau} \left(\nu K_{\tau}\left(\sqrt{\frac{\r}{d}}\,R\right)+\sqrt{d\r}K_{\tau+1}\left(\sqrt{\frac{\r}{d}}\,R\right)\right),
\end{array}\right.$$ thus, on $\p\O$, $U(y)=\Phi(|y|)$ satisfies the Robin boundary condition $$\label{2eq:4.2bis}
D\p_n U+\k U=0, \quad \text{ with } \k:=\frac{\mu\sqrt{d\r}K_{\tau+1}\left(\sqrt{\frac{\r}{d}}\,R\right)}{\nu K_{\tau}\left(\sqrt{\frac{\r}{d}}\,R\right)+\sqrt{d\r}K_{\tau+1}\left(\sqrt{\frac{\r}{d}}\,R\right)}>0.$$

Vice versa, if $U(y)=\Phi(|y|)$ is a positive solution of the first and last equations in which also satisfies , then, by inverting , we have that $V(y)=\Psi(|y|)$, where $\Psi$ is given by , satisfies the remaining equations of . Thus, the problem of existence and uniqueness of positive bounded solutions of with the specified symmetries is equivalent to the existence and uniqueness of positive radial solutions of the following boundary value problem $$\label{2eq:stationaryUD}
\left\{
\begin{array}{ll}
-D\, \D_y U=g(U), & \text{ in } B_{N-1}(R) \\
D\p_n U+\k U=0  & \text{ on } \p B_{N-1}(R).
\end{array}\right.$$

We begin with a non-existence result for problem , for which we assume a slightly more restrictive condition on $g$ than , precisely $$\label{eq:gKPP.medium}
g(0)=g(1)=0, \qquad 0< g(s)< g'(0)s \,\text{ for  $s\in(0,1)$,} \qquad g(s)<0 \,\text{ for  $s>1$}.$$

\[2pr:4.2\] Assume and that $$\label{2eq:nonexistenceD}
    \frac{g'(0)}{D}\leq\b_0^2$$ where $\b_0^2=\b_0^2(R,D)$ is the principal eigenvalue of the following Robin eigenvalue problem $$\label{2eq:linearD}
    \left\{
    \begin{array}{ll}
    -\D_y\phi=\b_0^2\,\phi, & \text{ in } B_{N-1}(R) \\
    D\p_n \phi+\k \phi=0  & \text{ on } \p B_{N-1}(R).
    \end{array}\right.$$ Then, problem has no positive radial solution.

Assume by contradiction that there exists a positive radial solution $U$.

First of all, we prove that $0<U<1$ in $\ov B$, where we denote $B:=B_{N-1}(R)$. Indeed, assume by contradiction that $\max U\geq 1$. Such a maximum has to be achieved in an interior point $y_0$, since, if it was achieved at the boundary, then $\p_n U\geq 0$ and the boundary condition in would not hold true. Moreover, if $U(y_0)=1$, the uniqueness of the Cauchy problems associated to the differential equation satisfied by the profile of $U$ would give $U\equiv 1$, against the boundary condition in again. Finally, if $U(y_0)>1$, then $-D\D_yU(y_0)\geq 0>g(U(y_0))$, again a contradiction. In a similar way, with the help of Hopf's lemma, we can exclude the existence of a minimum point where $U$ vanishes.

After these considerations, by taking a positive eigenfunction $\phi$ associated to $\b_0^2$, and using Green's identity in $B$, we obtain $$\label{2eq:4.7}
        -\int_{B}\D_y U\phi=-\int_{B} U\D_y\phi+\int_{\p B}\left(U\p_n\phi-\phi\p_n U\right)=\b_0^2\int_{B} U\phi.$$ On the other hand, by multiplying the differential equation of by $\phi$ and using and , we get $$-\int_{B}\D_y U\phi=\int_{B} \frac{g(U)}{D}\phi<\int_{B} \frac{g'(0)}{D}U\phi\leq\b_0^2\int_{B} U\phi,$$ yielding a contradiction with .

As an immediate consequence of Propositions \[2pr:4.1\] and \[2pr:4.2\], we obtain the extinction result of Theorem \[2th:section4.1\].

Before continuing, we give an useful characterization of the principal eigenvalue of problem in terms of the notation introduced in Section .

\[2pr:4.1bis\] The principal eigenvalue of problem satisfies $$\label{2eq:4.8}
        0<\b_0<\ov\b<\frac{j_\tau}{R},$$ where $\ov\b$ is the quantity defined in . Moreover, such an eigenvalue is given by the unique solution satisfying of the following relation: $$\label{2eq:4.8bis}
        \chi_u\left(\b_0\right)=\chi_v\left(\sqrt{\frac{\r}{d}}\right),$$ where $\chi_u$ and $\chi_v$ are the functions defined in . As a consequence, $$\label{2eq:4.8bisbis}
        \b_0>\un\b,$$ where $\un\b$ is the quantity introduced in and .

The unique solutions of the differential equation in which are defined in the whole $B_{N-1}(R)$ are $\phi(y)=\g|y|^{-\tau}J_{\tau}\left(\b_0|y|\right)$, with $\g>0$. In order $\phi$ to be positive, $\b_0 R$ has to lie in $(0,j_\tau)$. In addition, the boundary relation in is equivalent to $$\label{2eq:4.8ter}
            h_u\left(\b_0 R\right)=\frac{\mu R}{D}\frac{\sqrt{d\r}K_{\tau+1}\left(\sqrt{\frac{\r}{d}}\,R\right)}{\n K_{\tau}\left(\sqrt{\frac{\r}{d}}\,R\right)+\sqrt{d\r}K_{\tau+1}\left(\sqrt{\frac{\r}{d}}\,R\right)},$$ while, from , $h_u\left(\ov\b R\right)$ equals $\frac{\mu R}{D}$, which is greater than the right-hand side of . As a consequence, since $h_u(\b R)$ is increasing in $\b$, we have that $\b_0<\ov\b$ and, now, follows from with easy computations.

The previous results allow us to determine the long-time behavior of the solution of for large $N$, the other parameters of the problem being fixed, providing us with the proof of Theorem \[2th:section4.2\].

\[Proof of Theorem \[2th:section4.2\]\] From and , we have that $\b_0(N)\to+\infty$ as $N\to+\infty$. Thus, the conclusion follows by applying the extinction result in Theorem \[2th:section4.1\].

In addition, the characterization of $\b_0$ given in Proposition \[2pr:4.1bis\] allows us to establish the following properties of $\b_0$ as a function of $R$.

\[2pr:4.1ter\] The function $R\mapsto\b_0(R)$ is continuous and decreasing.

The continuity (actually differentiability) of this function follows directly from , since all the quantities appearing there are differentiable in $R$. Moreover, by explicitly indicating the dependence on $R$ in , we obtain $\chi_u\left(\b_0(R),R\right)=\chi_v\left(\sqrt{\frac{\r}{d}},R\right)$, and differentiating this relation with respect to $R$ gives $$\p_\b\chi_u\left(\b_0,R\right)\b_0'(R)+\p_R\chi_u\left(\b_0(R),R\right)=\p_R\chi_v\left(\sqrt{\frac{\r}{d}},R\right).$$ Thus, we obtain $\b_0'(R)<0$, since, as shown in Sections and \[1section4\], $\p_\b\chi_u>0$, $\p_R\chi_u>0$ and $\p_R\chi_v\leq 0$.

We are now able to give a positive lower bound for the solution of when condition holds true.

\[2pr:4.4\] Assume , , i.e. $$\label{2eq:existenceD}
    \frac{g'(0)}{D}>\b_0^2,$$ and let $(u,v)$ be the solution of starting with a non-negative, not equal to $(0,0)$, bounded initial datum. Then, there exists a positive bounded steady state $(U_1,V_1)$ which is $x$-independent, invariant by rotations about the axis $\{y=0\}$, and such that, locally uniformly in each domain, $$(U_1,V_1)\leq \liminf_{t\to+\infty}(u,v).$$

As for the lower bound in Proposition \[1pr:2.1\], we will show that, under assumption , it is possible to construct an arbitrarily small positive stationary subsolution $(\un u,\un v)$ of with compact support and which is invariant by rotations around $\{y=0\}$. The existence of $(U_1,V_1)$ and its symmetries then follow along the same lines.

If we are able to find a non-negative, not identically equal to zero, compactly supported pair $(u_*,v_*)$ satisfying $$\label{2eq:4.9}
        \left\{
        \begin{array}{ll}
        -D\D  u\leq (g'(0)-\t) u & \text{in ${{\mathbb{R}}}\times\{|y|<R\}$} \\
        -d\D  v=-\r  v & \text{in ${{\mathbb{R}}}\times\{|y|>R\}$} \\
        D\, \p_n  u(x,R) \leq \nu  v(x,R)-\mu  u(x,R) & \text{on ${{\mathbb{R}}}\times\{|y|=R\}$}\\
        -d\, \p_n  v(x,R) \leq \mu  u(x,R)-\nu  v(x,R) & \text{on ${{\mathbb{R}}}\times\{|y|=R\}$,} \\
        \end{array}\right.$$ with $\t>0$ sufficiently small, then, thanks to , there exists $\e_0>0$ such that $(\un{u},\un{v})=\e(u_*,v_*)$ is a compactly supported stationary subsolution of for $\e\in(0,\e_0)$.

Consider $R$ to be fixed, and let $\phi_1$ be a positive eigenfunction of with $R$ replaced by $R_1$, where $R_1$ is smaller than $R$ and sufficiently close to it so that $\b_0^2(R_1)<\frac{g'(0)}{D}$, which is possible thanks to and the continuity given by Proposition \[2pr:4.1ter\]. By considering the analytical extension of $\phi_1$ to $B_{N-1}(R)$ - which will still be denoted by $\phi_1$ -, and by taking $R_1$ closer to $R$ if necessary, we can assume without loss of generality that $\phi_1>0$ in $B_{N-1}(R)$.

We distinguish two cases: if $D\p_n\phi_1+\mu\phi_1\leq 0$ on $\p B_{N-1}(R)$, then

$$u_*:=\left\{\begin{array}{ll}
        \cos(\a x)\phi_1(y)&\text{if $(x,y)\in\left(-\frac{\pi}{2\a},\frac{\pi}{2\a}\right)\times B_{N-1}(R)$} \\
        0 &\text{otherwise,} 
        \end{array}\right. \qquad v_*:=0,$$

satisfies for $\a\sim 0$ and $\t\sim 0$ so that $\a^2+\b_0^2(R_1)<\frac{g'(0)-\t}{D}$.

Otherwise, if $$\label{2eq:4.10}
        D\p_n\phi_1+\mu\phi_1> 0 \quad \text{ on } \p B_{N-1}(R),$$ then $(u_*,v_*)$ can be taken as

$$\begin{gathered}
        u_*:=\left\{\begin{array}{ll}
        \cos(\a x)\phi_1\left( y\right)&\text{if $x\in\left(-\frac{\pi}{2\a},\frac{\pi}{2\a}\right)$, $|y|\in\left(0,R\right)$} \notag \\
        0 &\text{otherwise,} 
        \end{array}\right. \\   
        v_*:=\left\{\begin{array}{ll}
        \cos(\a x)\psi_1( y)&\text{if $x\in\left(-\frac{\pi}{2\a},\frac{\pi}{2\a}\right)$, $|y|\in\left(R,L\right)$} \\
        0 &\text{otherwise,} 
        \end{array}\right.
        \end{gathered}$$

for $\a\sim 0$ and $\t\sim 0$ satisfying $\a^2+\b_0^2(R_1)<\frac{g'(0)-\t}{D}$ as above, $L> R$ to be chosen, and where $\psi_1$ is the unique (radial) solution of $$\label{2eq:4.12}
        \left\{
        \begin{array}{ll}
        \D_y\psi=\left(\frac{\r}{d}+\a^2\right)\psi, & \text{ in } \{|y|\in(R,L)\} \\
        d\p_n \psi= D\p_n\phi_1  & \text{ on } \p B_{N-1}(R) \\
        \nu\psi=D\p_n\phi_1+\mu\phi_1 & \text{ on } \p B_{N-1}(R).
        \end{array}\right.$$ From Appendix \[sectionapp1\], we get that $\psi_1$ is given by $$\psi_1(y)=|y|^{-\tau}\left(C_1 I_{\tau}\left(\sqrt{\frac{\r}{d}+\a^2}\,|y|\right)+C_2 K_{\tau}\left(\sqrt{\frac{\r}{d}+\a^2}\,|y|\right)\right),$$ where $I_{\tau}$ is the modified Bessel function of first kind, and, in order to satisfy the initial conditions in , $C_1=C_1(\a)$ and $C_2=C_2(\a)$ are given by $$\begin{gathered}
        \label{2eq:4.13}
        C_1(\a)=
        \frac{\left(\nu K_\tau\left(\frac{r(\a)}{d}R\right)+r(\a)K_{\tau+1}\left(\frac{r(\a)}{d}R\right)\right)D\p_n\phi_1+\mu r(\a)K_{\tau+1}\left(\frac{r(\a)}{d}R\right)\phi_1}{R^{-\tau}\nu r(\a)\left(I_\tau\left(\frac{r(\a)}{d}R\right)K_{\tau+1}\left(\frac{r(\a)}{d}R\right)+K_\tau\left(\frac{r(\a)}{d}R\right)I_{\tau+1}\left(\frac{r(\a)}{d}R\right)\right)}, \\
        C_2(\a)=\frac{1}{K_{\tau+1}\left(\frac{r(\a)}{d}R\right)}\left(C_1 I_{\tau+1}\left(\frac{r(\a)}{d}R\right)-\frac{D\p_n\phi_1}{r(\a)R^{-\tau}}\right), \notag
        \end{gathered}$$ where we have set $r(\a):=\sqrt{d\r+d^2\a^2}$, and $\phi_1$ and $\p_n\phi_1$ are evaluated for $|y|=R$.

We claim that $C_1(0)<0$. Indeed, the denominator in is always positive, while, for $\a=0$, the numerator reduces to $$\left(\nu K_\tau\left(\sqrt{\frac{\r}{d}}R\right)+\sqrt{d\r}K_{\tau+1}\left(\sqrt{\frac{\r}{d}}R\right)\right)D\p_n\phi_1+\mu \sqrt{d\r}K_{\tau+1}\left(\sqrt{\frac{\r}{d}}R\right)\phi_1,$$ which is negative if and only if $$\label{2eq:4.14}
        D\p_n\phi_1+\k\phi_1<0.$$ Assume by contradiction that does not hold true. Then, by reasoning as in the proof of Proposition \[2pr:4.1bis\], we would have $$h_u\left(\b_0(R_1)R\right)\leq\frac{\mu R}{D}\frac{\sqrt{d\r}K_{\tau+1}\left(\sqrt{\frac{\r}{d}}\,R\right)}{\n K_{\tau}\left(\sqrt{\frac{\r}{d}}\,R\right)+\sqrt{d\r}K_{\tau+1}\left(\sqrt{\frac{\r}{d}}\,R\right)},$$ and, comparing with and using the monotonicity of $h_u(\b)$, this would give $\b_0(R_1)\leq\b_0(R)$, which is impossible by Proposition \[2pr:4.1ter\].

This proves our claim and and by reducing $\a$ if necessary, we have $C_1(\a)<0$ by continuity. As a consequence, since $\psi_1$ is positive on $\p B_{N-1}(R)$ thanks to and the last equation of , while it is negative for large $|y|$ since $I_{\tau}(r)\to+\infty$ and $K_{\tau}(r)\to 0$ for $r\to+\infty$, by taking $L$ as the first value of $|y|$ for which $\psi_1$ vanishes, all the desired conditions are satisfied.

The last step which is necessary to the study of the long-time behavior is the uniqueness of positive solutions of , which is obtained in the next proposition.

\[2pr:4.5\] Assume . Then, problem admits at most one positive solution.

Suppose by contradiction that there exists two positive solutions $U_1\neq U_2$ to . First of all, we prove that they cannot be ordered. Assume indeed, without loss of generality, that $U_1\leq U_2$ in $\ov{B_{N-1}(R)}$. Then, using Green's identity, and thanks to , we obtain the following contradiction $$\begin{gathered}
        0=\int_{B_{N-1}(R)}D \D U_2 U_1-D\D U_1 U_2 = \int_{B_{N-1}(R)}g(U_1)U_2-g(U_2)U_1= \\ =\int_{B_{N-1}(R)}\left(\frac{g(U_1)}{U_1}-\frac{g(U_2)}{U_2}\right)U_1U_2>0,
        \end{gathered}$$ the last inequality being strict since $U_1\neq U_2$. Thus, neither $U_1\leq U_2$, nor $U_1\geq U_2$. Nevertheless, $k U_1>U_2$ for $k$ large enough, while $k U_1<U_2$ for $k\sim 0$, since $U_1$ and $U_2$ are positive. Thus, there exists $$k_*:=\inf\{k>1: k U_1>U_2 \text{ in $\ov{B_{N-1}(R)}$}\}.$$ Observe that $k_*>1$, because we have proved that $U_1\geq U_2$ cannot hold true in $\ov{B_{N-1}(R)}$. Take now $M$ large enough so that the function $\tilde{g}(s):=g(s)+Ms$ is increasing for $s\in\left[0,k_*\max_{\ov{B_{N-1}(R)}}U_1\right]$, which is possible since $g$ is locally Lipschitz -, and set $W:=k_* U_1-U_2\geq 0$. It satisfies $$\left\{
        \begin{array}{ll}
        (-D\D_y+M)W=\tilde g(k_* U_1)-\tilde g(U_2)\geq 0, & \text{ in } B_{N-1}(R) \\
        D\p_n W+\k W=0  & \text{ on } \p B_{N-1}(R);
        \end{array}\right.$$ thus, since $W$ vanishes in (at least) one point by the definition of $k_*$, the strong maximum principle and Hopf's lemma imply that $W\equiv 0$, thus $U_2\equiv k_* U_1>U_1$, which is impossible.

As shown in the proof of Proposition \[2pr:4.2\], any positive solution of satisfies $0<U<1$. Thus, by examining the previous proof, we have that the conclusion of Proposition \[2pr:4.5\], as well as all the subsequent results, still holds true if we assume the monotonicity of only for $0<s<1$.

Moreover, it is immediate to get that implies that the large inequality in is actually strict, as required in .

We are finally able to conclude the study of the long-time behavior for the solutions of . The result is an immediate consequence of Propositions \[2pr:4.4\] and \[2pr:4.5\] and provides us with the invasion result of Theorem \[2th:section4.1\].

\[2th:4.6\] Assume , and . Then, the solution of starting from a non-negative, not identically equal to $(0,0)$, bounded initial datum satisfies $$\lim_{t\to+\infty}(u,v)=(U,V)$$ locally uniformly in each domain, where $(U,V)$ is the unique positive bounded steady state of which is $x$-independent and invariant by rotations about $\{y=0\}$.

When invasion occurs, we want to study the speed of propagation $c^*_m$ at which it takes place. For this reason, we assume without further reference , which, thanks to , and the monotonicity of $\chi_u(\b)$, can be equivalently written as $$\label{2eq:5.1}
        \frac{g'(0)}{D}>\b_0^2 \quad \iff \quad \sqrt{\frac{g'(0)}{D}}\geq\bar{\b} \quad \text{ or } \quad \begin{cases}
        \sqrt{\frac{g'(0)}{D}}<\bar{\b} \\
        \chi_v\left(\sqrt{\frac{\r}{d}}\right)<\chi_u\left(\sqrt{\frac{g'(0)}{D}}\right).
    \end{cases}$$

The construction of $c^*_m$ follows most of the lines as the one for problem , thus, we point out the only main differences. Exactly as in Section , we look for solutions of the linearization of problem around $(0,0)$ of the form , where $\d$ is implicitly given in terms of $\b$ though relation .

This leads to look for intersections between the curves $\Si_D$ defined in and $\tilde{\Si}_d(c)=\{(\b,\tilde{\a}^\pm_d(c,\b))\}$, where $$\tilde{\a}^\pm_d(c,\b):=\frac{c\pm\sqrt{c^2+4d\r-4d^2\d^2(\b)}}{2d}.$$ This curve is very similar to the one in , except for the fact that it is defined for all $c>0$ independently of the other parameters. Indeed, recalling and , we have $\d(\b_0)=\sqrt{\frac{\r}{d}}$; thus, for every $c>0$, there exists $\breve{\b}(c)\in(\b_0,\ov\b)$ such that $\tilde{\a}^\pm_d(c,\b)$ is defined for $\b\in[\un\b,\breve{\b}(c)]$. The function $c\mapsto\breve{\b}(c)$ is continuous, increasing and satisfies $$\lim_{c\da 0}\breve{\b}(c)=\b_0, \qquad \lim_{c\ua \infty}\breve{\b}(c)=\ov\b.$$ In analogy with the notation introduced above, we the denote by $\tilde{{\mathcal}{S}}_d(c)$ the region in the first quadrant of the $(\b,\a)$ plane lying between the curves $\tilde\a_d^\pm$.

Another major difference is that the construction performed in Proposition \[1pr:3.1\] of subsolutions for $c<c_f$ supported in ${{\mathbb{R}}}^N\setminus\ov\O$ cannot be repeated here, due to the fact that we are no longer considering a Fisher-KPP nonlinearity in such domain.

Nonetheless, we will now show that, analogously to what occurs in @T16 and @BRR4 for the road-field problems considered there, the regions $\mathcal{S}_D(c)$ and $\tilde{\mathcal{S}}_d(c)$ are separate for $c\sim 0$. Indeed, $\Si_D(0)$ is defined for $\b\geq\sqrt{\frac{g'(0)}{D}}$, while $\tilde\Si_d(0)$ is defined for for $\b\leq\b_0$, and $\b_0<\sqrt{\frac{g'(0)}{D}}$ thanks to .

As a consequence, $c^*_m$ can be constructed, as in Section , as the smallest value of $c>0$ for which $\Si_D(c)$ and $\tilde{\Si}_d(c)$ intersect, being tangent, for $\b>\un\b$. Observe that, if $D\geq d$, it is possible to show that the curves that touch, in the first quadrant, for the first time are $\tilde{\a}^-_d$ and $\a^+_D$, thus the first intersection necessarily occurs for $\b>\b_0>\un\b$. When $D<d$, instead, in order to guarantee the tangency for $\b>\un\b$, we have to restrict to $N\leq 5$ as in Section .

After these preliminaries, one can show that the desired supersolutions exist for all $c\geq c^*_m$, while compactly supported subsolutions of the problem with additional transport term can be obtained, for $c<c^*_m$, $c\sim c^*_m$, by considering a truncated problem with $v=0$ for $|y|=L$, $L\gg R$, and adapting the construction of Proposition \[1pr:3.2\]. In this way we have proved the last part of Theorem \[2th:section4.1\], asserting that $c^*_m$ is the asymptotic speed of propagation of problem .

Before concluding with the proof of the remaining qualitative properties of $c^*_m$ stated in Theorem \[2th:section4.2\], we present an auxiliary result in the spirit of Theorem \[1th:3.3\], which gives a sufficient condition for $c^*_m$ to be smaller than the Fisher-KPP speed $c_g$.

\[2pr:4.11\] If $$\label{2eq:6.8bisbis}
    (d-D)c_g-2d\un\b\leq 0, \quad \text{or} \quad \begin{cases}
    (d-D)c_g-2d\un\b> 0 \\
    \frac{d}{D}\left(1-C\right)^2\leq 2+\frac{\r}{g'(0)}-2C,
    \end{cases}$$ where $C:=\frac{\un\beta}{\sqrt{Dg'(0)}}$, then $c^*_m<c_g$.

Since $\tilde\a_d^-(c,\b_0)=0$ for every $c>0$, and $\a_D^-(c_g,\b)$ is a straight line passing through the points $\left(0,\frac{c_g}{2D}\right)$ and $\left(\sqrt{\frac{g'(0)}{D}},0\right)$, we have that, if the uppermost point of $\tilde\Si_d(c_g)$, which lies at $\b=\un\b$, lies not below $\a_D^-(c_g,\cdot)$, which is equivalent to $$\label{2eq:6.11}
        \tilde\a_d^+(c_g,\un\b)=\frac{c_g+\sqrt{c_g^2+4d\r}}{2d}\geq\frac{c_g}{2D}-\un\b=\a_D^-(c_g,\un\b),$$ then the tangency occurs before $c_g$, giving that $c^*_m<2\sqrt{Dg'(0)}$. Simple computations show that is equivalent to .

\[Proof of Theorem \[2th:section4.2\]\] Starting with the existence of $R_0$, by using Proposition \[2pr:4.1ter\], it will suffice to show that $$\label{2eq:6.3}
    \lim_{R\to 0}\b_0(R)=+\infty, \qquad \lim_{R\to+\infty}\b_0(R)=0$$ and to take $R_0$ as the unique value of $R$ for which $$\label{2eq:6.3bis}
    \b_0^2(R_0)=\frac{g'(0)}{D}.$$ The second limit in follows directly from . Regarding the first one, we obtain from and that $$\label{2eq:6.4}
    \b_0^2(R)=\frac{1}{R^2}k_u^2\left(\frac{\mu R}{D}\frac{\sqrt{d\r}K_{\tau+1}\left(\sqrt{\frac{\r}{d}}\,R\right)}{\n K_{\tau}\left(\sqrt{\frac{\r}{d}}\,R\right)+\sqrt{d\r}K_{\tau+1}\left(\sqrt{\frac{\r}{d}}\,R\right)}\right)=\frac{1}{R^2}k_u^2\left(\frac{\k R}{D}\right).$$ Since $\frac{\sqrt{d\r}K_{\tau+1}\left(\sqrt{\frac{\r}{d}}\,R\right)}{\n K_{\tau}\left(\sqrt{\frac{\r}{d}}\,R\right)+\sqrt{d\r}K_{\tau+1}\left(\sqrt{\frac{\r}{d}}\,R\right)}\in(0,1)$ is non-decreasing in $R$, thanks to , the argument of $k_u$ in converges linearly to $0$ as $R\to 0$. As a consequence, implies that $\lim_{R\to 0}\b_0^2(R)R\in(0,+\infty)$, and the desired result follows.

The monotonicity of $c^*_m(R)$ follows as in the proof of Theorem \[1th:3.3\]. Moreover, from the definition of $R_0$ given in , we have that $\b_0^2(R)\ua\frac{g'(0)}{D}$ as $R\da R_0$, thus, if $l:=\lim_{R\da R_0} c^*_m(R)$ was positive, then $$\lim_{R\da R_0}\breve{\b}(c^*_m(R),R)\geq\lim_{R\da R_0}\b_0(R)=\sqrt{\frac{g'(0)}{D}}>\hat{\b}(l),$$ and the curves $\Si_D$ and $\tilde\Si_d$ would intersect for $R\sim R_0$, against the construction of $c^*_m$.

The proof of the limit for $R\to+\infty$ follows also along the same lines of the one of Theorem \[1th:3.3\], with the only difference, here, that $\tilde\Si_d(c)$ converges to the vertical segment $$\left\{(0,\a):\frac{c-\sqrt{c^2+4d\r}}{2d}\leq \a\leq\frac{c+\sqrt{c^2+4d\r}}{2d} \right\}.$$

\[Proof of Theorem \[2th:section4.2\]\] To prove this result, we use the equivalent formulation of given by . First of all, we prove that the function $D\ov\b(D)^2$ is increasing. From , we have $$\label{2eq:6.6}
    \ov\b(D)=\frac{1}{R} k_u\left(\frac{\mu R}{D}\right),$$ thus $$\left(D\ov\b^2(D)\right)'=\frac{1}{R^2}k_u\left(\frac{\mu R}{D}\right)\left(k_u\left(\frac{\mu R}{D}\right)-2\frac{\mu R}{D}k_u'\left(\frac{\mu R}{D}\right)\right),$$ and $D\ov\b^2(D)$ is increasing if and only if $2sk_u'(s)<k_u(s)$ for every $s>0$, which, by setting $s=h_u(r)$, is equivalent to show that, for all $r\in(0,j_\tau)$, $$\label{2eq:6.7}
    2<r\frac{h_u'(r)}{h_u(r)}=2+r\left(\frac{J_{\tau+1}(r)}{J_{\tau}(r)}-\frac{J_{\tau+2}(r)}{J_{\tau+1}(r)}\right),$$ where, to get the last relation, we have used the definition of $h_u$ given in , and . It remains only to observe that the last addend in is positive thanks to .

From and , we obtain that $$\lim_{D\to\infty}D\ov\b^2(D)=\frac{\mu}{R}(N-1),$$ thus, if $\frac{\mu}{R}(N-1)\leq g'(0)$, we have that the first possibility in holds true for every $D>0$, and the first part of Theorem \[2th:section4.2\] is proved.

We consider now the case in which strict inequality holds true in : thanks to , holds true for large $D$, thus $c^*_m(D)<c_g$. Then, we reason as in the proof of Theorem \[1th:3.3\]: if we assume by contradiction that $c^*_m(D)$ is bounded, we obtain that the tangency would occur between $\tilde\a_d^-$ and $\a_D^+$. Moreover, by combining and , we obtain, for $D\sim\infty$, $$\label{2eq:6.14}
        \b_0^2(D)=\frac{\k(N-1)}{R}\frac{1}{D}-\frac{N-1}{N+1}\frac{\k^2}{D^2}+o\left(D^{-2}\right),$$ thus, since strict inequality holds true in , we have that $\lim_{D\to\infty}D\b_0^2(D)<g'(0)$, and the curves $\tilde\a_d^-\left(c^*_m(D),\frac{\b'}{\sqrt{D}}\right)$ and $\a_D^+\left(c^*_m(D),\frac{\b'}{\sqrt{D}}\right)$ would be separate apart for large $D$ (see the details in the proof of Theorem \[1th:3.3\] and observe that we have performed the change of variable $\b'=\b\sqrt{D}$ as in that proof). This contradicts the construction of $c^*_m$ and shows that $c^*_m(D)$ is unbounded as $D\to+\infty$.

To conclude this case, we observe that the rate of convergence to $+\infty$ follows again as in the proof of Theorem \[1th:3.3\].

Passing to the case in which equality holds true in . As already remarked in the previous cases, for large $D$ and every fixed $c>0$ we have $\frac{c}{2D}<\frac{c}{2d}$. Thus, for $c=c^*_m$, the tangency occurs between $\a_D^+$ and $\tilde\a_d^-$ and $\hat\b(c^*_m(D),D)>\b_0(D)$, because otherwise the curves will be secant. This last relation and the definition of $\hat\b(c)$ given in yield to $$c^{*2}_m(D)=4D(g'(0)-D\hat\b^2(c^*_m(D),D))<4D\left(g'(0)-D\b_0^2(D)\right).$$ The desired result now follows by taking the $\limsup$ for $D\to+\infty$ in the last relation, using and the fact that now, since equality holds true in , we have $\frac{\k(N-1)}{R}=g'(0)$.

Finally, if $\frac{\mu}{R}(N-1)> g'(0)$, since $k_u$ is bounded, we have that $\lim_{D\da0}D\ov\b^2(D)=0$ and there exists $D_1>0$ such that $$\label{2eq:6.8}
    \sqrt{\frac{g'(0)}{D}}\geq\ov\b(D) \text{ for $D\leq D_1$}, \quad \sqrt{\frac{g'(0)}{D}}<\ov\b(D) \text{ for $D> D_1$}, \quad \lim_{D\da D_1}\sqrt{\frac{g'(0)}{D}}=\ov\b(D_1).$$ As a consequence, holds true for $D\leq D_1$ and, for $D>D_1$ if and only if $\chi_v\left(\sqrt{\frac{\r}{d}}\right)<\chi_u\left(\sqrt{\frac{g'(0)}{D}}\right)$, which reads $$\sqrt{d\r}\frac{K_{\tau+1}\left(\sqrt{\frac{\rho}{d}}R\right)}{K_{\tau}\left(\sqrt{\frac{\rho}{d}}R\right)}<\frac{\nu g'(0)R \frac{J_{\tau+1}(s(D))}{s(D)J_{\tau}(s(D))}}{\mu-g'(0)R \frac{J_{\tau+1}(s(D))}{s(D)J_{\tau}(s(D))}},$$ where we have set $s(D):=\sqrt{\frac{g'(0)}{D}}R$. The right-hand side is decreasing in $D$, since so is $s(D)$ and thanks to Lemma \[le:A.1\]. Moreover, from the last relation in , it converges to $+\infty$ as $D\da D_1$ and to $\frac{\nu g'(0)R}{\mu(N-1)-g'(0)R}$ as $D\to\infty$, thanks to . It is easy to see now that $D_0$ is given by the unique value of $D$ for which $$\label{2eq:6.8bis}
    \chi_v\left(\sqrt{\frac{\r}{d}}\right)=\chi_u\left(\sqrt{\frac{g'(0)}{D_0}}\right).$$ The limit of $c^*_m(D)$ for $D\ua D_0$ follows by reasoning as in Theorem \[2th:section4.2\], since, from and , we have that $\b_0^2(D)\ua\frac{g'(0)}{D_0}$ as $D\ua D_0$.

Theorem \[2th:section4.2\] implies that, when does not hold true, $c^*_m(D)$ is decreasing for $D<D_0$, $D\sim~D_0$, giving another example, in addition to the one commented in Remark \[1re:4.2\] for problem , in which the asymptotic speed of propagation is not increasing with respect to $D$. We point out that, from the construction performed in @BRR4 (with $g\equiv 0$), it is possible to show that, instead, the speed of propagation of the road-field problem corresponding to is increasing.

Some properties of Bessel functions {#sectionapp1}
===================================

In this Appendix we gather several facts on Bessel functions of the first kind and modified Bessel functions of the first and second kind that we use used in this work. When no reference is explicitly given, we send the interested reader to the complete monograph @W. We provide only the proofs of the results for which we have not been able to find any reference, since we think that they can be of independent interest.

The Bessel function of the first kind $J_\tau(r)$, $\tau>-1$, is a solution which is defined for $r\geq 0$ of the differential equation $$\label{1eq:A.-1}
    r^2 w''(r)+rw'(r)+(r^2-\tau^2)w(r)=0.$$ It has a first positive zero which will be denoted by $j_\tau$, it is positive for $r\in(0,j_\tau)$, and we have (see @W [p. 508, (3)]) that $$\label{1eq:A.0}
\tau\mapsto j_\tau \quad \text{ is increasing}.$$

The following asymptotic expansion holds true (see @W [pp. 40, 41]) $$\label{1eq:A.1}
    J_{\tau}(r)=\left(\frac{r}{2}\right)^\tau\left(\frac{1}{\G(\tau+1)}-\frac{r^2}{4\G(\tau+2)}+O(r^4)\right) \quad \text{ for $r\sim 0$},$$ where $\G$ is the Gamma function, as well as the following relation for the derivative (see @W [p. 45]) $$\label{1eq:A.2}
    (r^{-\tau}J_\tau(r))'=-r^{-\tau}J_{\tau+1}(r).$$ Moreover, we have the following monotonicity property for the quotient of Bessel functions of the first kind.

\[le:A.1\] For every $\e>0$, $\a\geq -\e$ and $\tau>-1$, the function $r \mapsto r^\a\frac{J_{\tau+\e}(r)}{J_{\tau}(r)}$ is increasing for $r\in(0,j_{\tau})$.

By using , we obtain $$\begin{gathered}
\left(r^\a\frac{J_{\tau+\e}(r)}{J_{\tau}(r)}\right)'=\left(r^{\a+\e}\frac{r^{-(\tau+\e)}J_{\tau+\e}(r)}{r^{-\tau}J_{\tau}(r)}\right)'=(\a+\e)r^{\a-1}\frac{J_{\tau+\e}(r)}{J_{\tau}(r)}+ \\
            +r^{\a}\frac{J_{\tau+\e}(r)J_{\tau+1}(r)-J_{\tau+\e+1}(r)J_{\tau}(r)}{J_{\tau}^2(r)},
        \end{gathered}$$ which, in the considered range of $r$, is the sum of non-negative terms, since $J_\tau$ and $J_{\tau+\e}$ are positive by , and since, as shown in @B [Remark 2, p. 289], the function $$\label{1eq:A.2bis}
        \tau\mapsto\frac{J_{\tau+1}}{J_{\tau}} \text{ is decreasing for $r\in(0,j_\tau)$ and $\tau>-1$,}$$ thus $\frac{J_{\tau+1}(r)}{J_{\tau}(r)}>\frac{J_{\tau+\e+1}(r)}{J_{\tau+\e}(r)}$.

Thanks to the previous lemma, the function $$\label{1eq:A.3}
    \begin{array}{rl}
    h_u(r):(0,j_\tau)&\to(0,\infty) \\
    r&\mapsto \displaystyle{r\frac{J_{\tau+1}(r)}{J_{\tau}(r)}}
    \end{array}$$ is increasing, invertible and, by , it satisfies $$\label{1eq:A.6}
        h_u(r)=\frac{r^2}{2(\tau+1)}\left(1+\frac{r^2}{4(\tau+1)(\tau+2)}+o(r^2)\right) \quad \text{as $r\sim 0$},$$ while its inverse, which will be denoted by $k_u(r)$ and is also increasing, thanks to the previous relation satisfies $$\label{1eq:A.7}
    k_u^2(r)=2(\tau+1)r-\frac{\tau+1}{\tau+2}r^2+o(r^2), \quad \text{ for $r\sim 0$}.$$

Another monotonicity property related to quotients of Bessel functions of the first kind is given in the following proposition.

\[pr:A.3\] For $\tau>-1$ and $r\in(0,j_{\tau})$, the function $\frac{h_u'(r)}{r}$ is increasing.

Using , direct computations give $$\begin{gathered}
            \frac{h_u'(r)}{r}=2\frac{J_{\tau+1}}{rJ_{\tau}}+\frac{J_{\tau+1}^2-J_{\tau+2}J_{\tau}}{J_{\tau}^2}=2\frac{J_{\tau+1}}{rJ_{\tau}}+\sum_{n\in{{\mathbb{N}}}}4(\tau+2+2n)\left(\frac{J_{\tau+2+2n}}{rJ_{\tau}}\right)^2,
            \end{gathered}$$ where the last equality comes from Lommel's formula (see @W [p. 152]). We then conclude by observing that the first summand in the right-hand side is increasing thanks to Lemma \[le:A.1\] applied with $\e=-\a=1$, and the same occurs for each term of the series thanks to Lemma \[le:A.1\] applied with $\e=2+2n$ and $\a=-1$.

The last property that we recall is the following asymptotic expansion for large orders (see @AS [$\S$ 9.3.1]): $$\label{1eq:A.8}
        J_{\tau}(r)\sim\frac{1}{\sqrt{2\pi\tau}}\left(\frac{e\, r}{2\tau}\right)^{\tau} \quad \text{as $\tau\to+\infty$.}$$

Passing to the modified Bessel function of the second kind $K_{\tau}(r)$, it is a solution of the differential equation $$\label{2eq:A.1}
    r^2 w''(r)+rw'(r)-(r^2+\tau^2)w(r)=0$$ which is positive and defined for all $r>0$, and decays to $0$ as $r\to+\infty$, since (see @W [p. 202]) $$\label{2eq:A.2}
    \lim_{r\to+\infty}K_{\tau}(r)\left(\frac{\pi}{2r} \right)^{-\frac{1}{2}}e^{r}=1.$$ On the other hand, the following asymptotic expansions, where we take $N\in{{\mathbb{N}}}$, $N\geq 2$ and $\tau=\frac{N-3}{2}$, hold true for $r>0$, $r\sim 0$ (see @W [p. 80]): $$\label{2eq:A.3}
    K_{\tau}(r)\sim\begin{cases}
        r^{-\frac{1}{2}}\sqrt{\frac{\pi}{2}}\left(1-r+\frac{r^2}{2}+O(r^3) \right) & \text{if $N=2,4$} \\
        C-\log(r)+\frac{r^2}{4}\left(1+C-\log(r) \right)+o(r^3) & \text{if $N=3$} \\
        r^{-1}\left(1-\frac{r^2}{4}\left(1+2C-2\log(r)\right) \right) +o(r^3)& \text{if $N=5$} \\
        r^{-\tau}\G(\tau)2^{\tau-1}\left(1-\frac{r^2}{4(\tau-1)}+o(r^3) \right) & \text{if $N\geq 6$}.
    \end{cases}$$ In addition, we have the following relation (see @W [p. 79]) $$\label{2eq:A.4}
    (r^{-\tau}K_\tau(r))'=-r^{-\tau}K_{\tau+1}(r),$$ and, thanks to @W [p. 79] and @IM [Lemma 2.4] respectively, we have, for all $r>0$, $$\label{2eq:A.5}
  \frac{K_{1/2}(r)}{K_{-1/2}(r)}=1,  \qquad r\mapsto\frac{K_{\tau+1}(r)}{K_\tau(r)} \text{ is decreasing for $\tau>-1/2$}.$$ Moreover, $$\label{2eq:A.6}
h_v(r):= \displaystyle{r\frac{K_{\tau+1}(r)}{K_{\tau}(r)}} \text{ is increasing for $r>0$},$$ since $K'_{\tau}(r)=-\frac{rK_{\tau-1}(r)+\tau K_{\tau}(r)}{r}$ (see @W [p. 79]), thus $$\left(r\frac{K_{\tau+1}(r)}{K_{\tau}(r)}\right)'=r\frac{K_{\tau+1}(r)K_{\tau-1}(r)-K_{\tau}(r)^2}{K_{\tau}(r)^2},$$ and the numerator in the last expression is positive (see @IM [(1.7)]).

By using , and recalling that $\G\left(\frac{3}{2}\right)=\frac{\sqrt{\pi}}{2}$, we have that $$\label{2eq:A.7}
\lim_{r\da0}h_v(r)=\max\{0,N-3\}.$$

Finally, we briefly recall some properties of the modified Bessel function of first kind $I_{\tau}(r)$, which is a solution of that is linearly independent with respect to $K_{\tau}(r)$, is positive for all $r>0$, increasing, since $$\label{3eq:A.2}
    (r^{-\tau}I_\tau(r))'=r^{-\tau}I_{\tau+1}(r),$$ (see @W [p. 79]), and it satisfies (see @W [p. 203]) $$\label{3eq:A.3}
    \lim_{r\to+\infty}I_{\tau}(r)=+\infty.$$

Acknowlegments {#acknowlegments .unnumbered}
==============

This work has been supported by: the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement number 321186 - ReaDi Reaction-Diffusion Equations, Propagation and Modelling"; the Institute of Complex Systems of Paris Île-de-France under DIM 2014 Diffusion heterogeneities in different spatial dimensions and applications to ecology and medicine"; the Project NONLOCAL (ANR-14-CE25-0013) founded by the French National Research Agency; and the Spanish Ministry of Economy, Industry and Competitiveness through project MTM2015-65899-P and contract Juan de la Cierva Incorporación IJCI-2015-25084.

<span>10</span> M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, vol. 55, Tenth Printing, Washington, 1972.

D.G. Aronson, H.F. Weinberger, *Multidimensional nonlinear diffusion arising in population genetics*, Adv. in Math. **30** (1978), 33–76.

Á. Baricz, *Functional inequalities involving Bessel and modified Bessel functions of the first kind*, Expo. Math. **26** (2008), 279–293.

H. Berestycki, F. Hamel, L. Roques, *Analysis of the periodically fragmented environment model. <span>I</span>. <span>S</span>pecies persistence*, J. Math. Biol. **51** (2005), 75–113.

H. Berestycki, G. Nadin, *Asymptotic spreading for general heterogeneous Fisher-KPP type equations* (2017), [preprint](http://lodel.ehess.fr/cams/docannexe.php?id=1344).

H. Berestycki, J.-M. Roquejoffre, L. Rossi, *The influence of a line with fast diffusion in Fisher-KPP propagation*, J. Math. Biol. **66** (2013), 743–766. H. Berestycki, J.-M. Roquejoffre, L. Rossi, *Fisher-KPP propagation in the presence of a line: further effects*, Nonlinearity **29** (2013), 2623–2640. H. Berestycki, J.-M. Roquejoffre, L. Rossi, *The shape of expansion induced by a line with fast diffusion in Fisher-KPP equations*, **343** Comm. Math. Phys. (2016), 207–232.

H. Berestycki, J.-M. Roquejoffre, L. Rossi, *Travelling waves, spreading and extinction for <span>F</span>isher-<span>KPP</span> propagation driven by a line with fast diffusion*, Nonlinear Anal. **137** (2016), 171–189.

A. Morris, L. Börger, E. Crooks, *Individual variability in dispersal and invasion speed*, [arXiv:1612.06768](https://arxiv.org/abs/1612.06768).

T. de Camino-Beck, M. A. Lewis, *Invasion with stage-structured coupled map lattices: Application to the spread of scentless chamomile*, Ecol. Modell. **220** (2009) 3394–3403.

N.R. Faria et al., *The early spread and epidemic ignition of HIV-1 in human populations*, Science **346** (2014), 56–61.

K. Fellner, E. Latos, B.Q. Tang *Well-posedness and exponential equilibration of a volume-surface reaction-diffusion system with nonlinear boundary coupling*, Ann. Inst. H. Poincaré Anal. Non Linéaire **35** (2018), 643–673. R.A. Fisher, *The wave of advantage of advantageous genes*, Annals of Eugenics **7** (1937) 355–369. M. Holzer, *Anomalous spreading in a system of coupled <span>F</span>isher-<span>KPP</span> equations*, Phys. D **270** (2014), 1–10.

M.E.H. Ismail, M.E. Muldoon, *Monotonicity of the zeros of a cross-product of Bessel functions*, SIAM J. Math. Anal. **9** (1978), 759–767. A.N. Kolmogorov, I.G. Petrovskii, N.S. Piskunov, *Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à  un problème biologique*, Bull. Univ. État Moscou, Sér. Intern. A **1** (1937), 1–25.

H. Li, X. Wang, *Using effective boundary conditions to model fast diffusion on a road in a large field*, Nonlinearity **30** (2017), 3853–3894.

A. Madzvamuse, A.H.W. Chung, C. Venkataraman, *Stability analysis and simulations of coupled bulk-surface reaction-diffusion systems*, Proc. A. **471** (2015), 18 pp. H.W. McKenzie, E.H. Merrill, R.J. Spiteri, M.A. Lewis, *How linear features alter predator movement and the functional response*, Interface focus, **2** (2012), 205–216.

L. Roques, J.-P. Rossi, H. Berestycki, J. Rousselet, J. Garnier, J.-M. Roquejoffre, L. Rossi, S. Soubeyrand, C. Robinet, Modeling the spatio-temporal dynamics of the pine processionary moth, in *Processionary Moths and Climate Change: An Update*, Springer, 2015.

L. Rossi, A. Tellini, E. Valdinoci, *The effect on Fisher-KPP propagation in a cylinder with fast diffusion on the boundary*, SIAM J. Math. Anal. **49** (2017), 4595–4624.

A. Tellini, *Propagation speed in a strip bounded by a line with different diffusion*, J. Differential Equations **260** (2016), 5956–5986.

G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1944.

H. F. Weinberger, M. A. Lewis, B. Li, *Anomalous spreading speeds of cooperative recursion systems.*, J. Math. Biol. **55** (2007), 207–222.

[^1]: burnsandrea@example.com

[^2]: burnsandrea@example.com

[^3]: burnsandrea@example.com

{'source': 'arXiv_out/1903'}