Traveling waves for degenerate diffusive equations on networks
Andrea Corli Lorenzo Di Ruvo Luisa Malaguti Massimiliano Rosini























Traveling waves for degenerate diffusive equations on networks
    Andrea Corli

Department of Mathematics and Computer Science

University of Ferrara, I-44121 Italy


Lorenzo di Ruvo

Department of Sciences and Methods for Engineering

University of Modena and Reggio Emilia, I-42122 Italy



Luisa Malaguti

Department of Sciences and Methods for Engineering

University of Modena and Reggio Emilia, I-42122 Italy 



Massimiliano D. Rosini

Department of Mathematics

Maria Curie-Skłodowska-University, PL-20031 Poland
    December 30, 2023
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In this paper we consider a scalar parabolic equation on a star graph; the model is quite general but what we have in mind is the description of traffic flows at a crossroad. In particular, we do not necessarily require the continuity of the unknown function at the node of the graph and, moreover, the diffusivity can be degenerate. Our main result concerns a necessary and sufficient algebraic condition for the existence of traveling waves in the graph. We also study in great detail some examples corresponding to quadratic and logarithmic flux functions, for different diffusivities, to which our results apply.
 


AMS Subject Classification: 35K65; 35C07, 35K55, 35K57

Keywords: Parabolic equations, wavefront solutions, traffic flow on networks.




§ INTRODUCTION

Partial differential equations on networks have been considered in the last years by several authors, in particular in the parabolic case; we quote for instance <cit.>. According to the modeling in consideration and to the type of equations on the edges of the underlying graph, different conditions at the nodes are imposed. In most of the cases, precise results of existence of solutions are given, even for rather complicated networks.

In this paper, the main example we have in mind comes from traffic modeling, where the network is constituted by a crossroad connecting m incoming roads with n outgoing roads; the traffic in each road is modeled by the scalar diffusive equation

    ρ_h,t + f_h(ρ_h)_x = (D_h(ρ_h)ρ_h,x)_x,      h = 1,…,m+n,

where t denotes time and x the position along the road.
In this case ρ_h is a vehicle density; about the diffusivity D_h(ρ_h)≥0 we do not exclude that it may vanish at some points. System (<ref>) is completed by a condition of flux conservation at the crossroad, which implies the conservation of the total number of cars. Such a model is derived from the famous Lighthill-Whitham-Richards equation <cit.>. We refer to <cit.> for several motivations about the introduction of (possibly degenerate) diffusion in traffic flows and in the close field of crowds dynamics. We also refer to the recent books <cit.> for more information on the related hyperbolic modeling.

We focus on a special class of solutions to (<ref>), namely, traveling waves. In the case of a single road, traveling waves are considered, for instance, in <cit.>; in the case of a second-order model without diffusion but including a relaxation term, we refer to <cit.>; for a possibly degenerate diffusion function and in presence of a source term, detailed results are given in <cit.>. In the case of a network, the papers dealing with this subject, to the best of our knowledge, are limited to <cit.> for the semilinear diffusive case and to  <cit.> for the case of a dispersive equation. In these papers, as in most modeling of diffusive or dispersive partial differential equations on networks, both the continuity of the unknown functions and the Kirchhoff condition (or variants of it) are imposed at the nodes. We emphasize that while the classical Kirchhoff condition implies the conservation of the flow and then that of the mass, some variants of this condition are dissipative and, then, imply none of the conservations above. While these assumptions are natural when dealing with heat or fluid flows, they are much less justified in the case of traffic modeling, where the density must be allowed to jump at the node while the conservation of the mass must always hold. Moreover, they impose rather strong conditions on the existence of the profiles, which often amount to proportionality assumptions on the parameters in play.

In this paper we only require the conservation of the (parabolic) flux at the node, as in <cit.>; differently from that paper and the other ones quoted above, we do not impose the continuity condition. A strong motivation for dropping this condition comes from the hyperbolic modeling <cit.>; nevertheless, we show how our results simplify when such a condition is required. In particular, in Sections <ref> and <ref> we provide explicit conditions for traveling wave solutions which do not satisfy the continuity condition; in some other cases, such a condition is indeed always satisfied. Our main results are essentially of algebraic nature and concern conditions about the end states, flux functions, diffusivities and other parameters which give rise to a traveling wave moving in the network. 

Here follows a plan of the paper. In Section <ref> we introduce the model and give some basic definitions; for simplicity we only focus on the case of a star graph. Section <ref> deals with a general existence result in the case of a single equation; its proof is provided in Appendix <ref>. Section <ref> contains our main theoretical results about traveling waves in a network. In that section we characterize both  stationary/non-stationary and degenerate/non-degenerate waves; in particular, Theorem <ref> contains an important necessary and sufficient condition that we exploit in the following sections. Section <ref> focus on the continuity condition; in this case the conditions for the existence of traveling wave solutions are much stricter than in the previous case. Detailed applications of these results are provided in Sections <ref> for quadratic fluxes and in Section <ref> for logarithmic fluxes; in particular, in subsection <ref> and in the whole Section <ref> the diffusivity is as in <cit.>. For simplicity, we only deal there with the case of a single ingoing road but we consider both constant and degenerate diffusivities.





§ THE MODEL

In terms of graph theory, we consider a semi-infinite star-graph with m incoming and n outgoing edges; this means that the incidence vector d∈^m+n has components d_i=1 for i∈𝖨≐{ 1, …, m } and d_j=-1 for j∈𝖩≐{ m+1, …, m+n }. We also denote 𝖧≐{ 1,…, m+n } and refer to Figure <ref>. For simplicity, having in mind the example in the Introduction, we always refer to the graph as the network, to the node as the crossroad and to the edges as the roads. Then, incoming roads are parametrized by x ∈_-≐(-∞,0] and numbered by the index i, outgoing roads by x ∈_+≐[0,∞) and j; the crossroad is located at x=0 for both parameterizations. We denote the generic road by Ω_h for h ∈𝖧; then Ω_i ≐_- for i ∈𝖨 and Ω_j ≐_+ for j ∈𝖩. The network is defined as 𝒩≐∏_h∈𝖧Ω_h.







Following the above analogy, we understand the unknown functions ρ_h as vehicular densities in the roads Ω_h, h∈𝖧; ρ_h ranges in [0,_h], where _h is the maximal density in the road Ω_h. Without loss of generality we assume that _h=1 for every h∈𝖧; the general case is easily recovered by a change of variables and modifying (<ref>)-(<ref>) below for a multiplicative constant. With a slight abuse of notation we denote ρ≐ (ρ_1, …, ρ_m+n): ×𝒩→ [0,1]^m+n understanding that ρ(t,x_1,…, x_m+n)=(ρ_1(t,x_1),…,ρ_m+n(t,x_m+n)).

For each road we assign the functions f_h, the hyperbolic flux, and D_h, the diffusivity; we assume for every h ∈𝖧



  (f) f_h∈𝐂^1([0,1];_+) is strictly concave with f_h(0)=f_h(1)=0;


  (D) D_h∈𝐂^1([0,1];_+) and D_h(ρ)>0 for any ρ∈ (0,1).


We emphasize that in (D) we can possibly have either D_h(0)=0 or D_h(1)=0, or even both possibilities at the same time. The evolution of the flow is described by the equations

    ρ_h,t + f_h(ρ_h)_x = (D_h(ρ_h)ρ_h,x)_x,      (t,x) ∈ℝ×Ω_h,  h ∈𝖧.

Assumption (f) is standard when dealing with traffic flows <cit.>. More precisely, in that case f_h(ρ_h)=ρ_h   v_h(ρ_h), where v_h is the velocity.
Then, assumption (f) is satisfied if, for instance, v_h∈𝐂^2([0,1];_+) is either linear or strictly concave, decreasing and satisfying v_h(1) = 0, see <cit.>.
The prototype of such a velocity satisfying (f) is v_h(ρ) = V_h (1-ρ) with V_h > 0, which was introduced in <cit.>; another example is given in <cit.>.
The simplest model for the diffusivity is then D_h(ρ_h) = -δ_hρ_h   v_h'(ρ_h), where δ_h is an anticipation length <cit.>.

The coupling among the differential equations in (<ref>) occurs by means of suitable conditions at the crossroad. In this paper, having in mind the previous example, we impose a condition on the conservation of the total flow at the crossroad, see <cit.>; in turn, this implies the conservation of the mass. More precisely, we define the parabolic flux by

    F_h(ρ_h,ρ_h,x) ≐ f_h(ρ_h) - D_h(ρ_h)  ρ_h,x

and require

    F_j(ρ_j(t,0^+), ρ_j,x(t,0^+))
    = ∑_i ∈𝖨α_i,j  F_i(ρ_i(t,0^-), ρ_i,x(t,0^-))
        for a.e. t∈, j ∈𝖩,

for given constant coefficients α_i,j∈ (0, 1] satisfying

    ∑_j ∈𝖩α_i,j = 1,      i ∈𝖨.

Conditions (<ref>) and (<ref>) imply

    ∑_j ∈𝖩 F_j(ρ_j(t,0^+), ρ_j,x(t,0^+))
    =
    ∑_i ∈𝖨 F_i(ρ_i(t,0^-), ρ_i,x(t,0^-))
        for a.e. t∈,

which is the conservation of the total flow at the crossroad. Conditions (<ref>) and (<ref>) deserve some comments. First, by no means they imply

    ρ_i(t,0^-) = ρ_j(t,0^+),      t ∈,   (i,j)  ∈𝖨×𝖩.

Condition (<ref>) is largely used, together with some Kirchhoff conditions, when dealing with parabolic equations in networks and takes the name of continuity condition. Second, above we assumed α_i,j>0 for every i and j. The case when α_i,j=0 for some i and j would take into account the possibility that some outgoing j roads are not allowed to vehicles coming from some incoming i roads; this could be the case, for instance, if only trucks are allowed in road i but only cars are allowed in road j. For simplicity, we do not consider this possibility. Third, we notice that assumption (<ref>) destroys the symmetry of condition (<ref>); indeed, with reference to the example of traffic flow, the loss of symmetry is due to the fact that all velocities v_h are positive.

Then, we are faced with the system of equations (<ref>) that are coupled through (<ref>), with the α_i,j satisfying (<ref>). Solutions to (<ref>)-(<ref>) are meant in the weak sense, namely ρ_h∈𝐂^1(×Ω_h;[0,1]) a.e.; see also <cit.> for an analogous definition in the hyperbolic case. We do not impose any initial condition because we only consider traveling waves, which are introduced in the next sections.





§ TRAVELING WAVES FOR A SINGLE EQUATION


In this section we briefly remind some definitions and results about traveling waves <cit.> for the single equation

    ρ_h,t + f_h(ρ_h)_x = (D_h(ρ_h)ρ_h,x)_x,      (t,x) ∈ℝ×Ω_h,

where we keep for future reference the index h. Equation (<ref>) has no source terms (differently from <cit.>) and then any constant is a solution; for simplicity we discard constant solutions in the following analysis.

 A weak solution ρ_h(t,x)  to (<ref>) is a traveling-wave solution of (<ref>) if ρ_h(t,x) = ϕ_h(x-c_h t) for (t,x) ∈ℝ×Ω_h, for a non-constant profile ϕ_h:→[0,1] and speed c_h ∈.


This definition coincides with that given in <cit.> because we are considering non-constant profiles. The profile must satisfy the equation

    [F_h(ϕ_h,ϕ_h')- c_h ϕ_h ]'=0,

namely,

    (D_h(ϕ_h)ϕ_h')' - g_h'(ϕ_h)  ϕ_h'=0,

in the weak sense, where

    g_h(ρ) ≐ f_h(ρ) - c_h  ρ

is the reduced flux, see  <ref>.





This means that ϕ_h ∈0(ℝ; [0,1]), D_h(ϕ_h)  ϕ_h'∈1(ℝ;ℝ) and

    ∫_ℝ[D_h(ϕ_h(ξ))ϕ_h'(ξ) - g_h(ϕ_h(ξ)) ] ψ'(ξ)   dξ =0,

for every ψ∈∞(ℝ;ℝ). Equation (<ref>) is  coupled with the limit conditions

    ϕ_h(±∞) = ℓ_h^±,

for ℓ_h^±∈[0,1]. Clearly, solutions to (<ref>)-(<ref>) are determined up to a shift. We define

    I_h ≐{ξ∈ℝ  :  ℓ_h^-<ϕ_h(ξ)<ℓ_h^+}.


The existence of profiles is a well-established result <cit.>; nevertheless, we state for completeness the following theorem, where we point out the qualitative properties of these fronts. The proof is deferred to Appendix <ref>.


Assume (f) and (D). Equation (<ref>) admits a traveling-wave solution ρ_h with profile ϕ_h satisfying (<ref>) if and only if

    0≤ℓ_h^- < ℓ_h^+≤1       and       c_h = f_h(ℓ_h^+) - f_h(ℓ_h^-)ℓ_h^+ - ℓ_h^-.

We have that ϕ_h ∈𝐂^2(I_h; (ℓ_h^-, ℓ_h^+)) is unique (up to shifts) and ϕ_h'(ξ)>0 for ξ∈ I_h; moreover, the following holds true.



  (i) D_h(0)=0=ℓ_h^- if and only if there exists ν_h^- ∈ℝ such that I_h ⊆ (ν_h^-,∞) and ϕ_h(ξ)=0 for ξ≤ν_h^-.
In this case

    lim_ξ↓ν_h^-ϕ_h'(ξ)=ℓ_h^+f_h'(0)-f_h(ℓ_h^+)/ℓ_h^+D_h'(0)   if  D_h'(0)>0,
    ∞   if  D_h'(0)=0,
    
       lim_ξ↓ν_h^-D_h(ϕ_h(ξ))ϕ_h'(ξ) = 0.



  (ii) D_h(1)=0=1-ℓ_h^+ if and only if there exists ν_h^+ ∈ℝ such that I_h ⊆ (-∞,ν_h^+) and ϕ_h(ξ)=1 for ξ≥ν_h^+.
In this case

    lim_ξ↑ν_h^+ϕ_h'(ξ)=(1-ℓ_h^-)f_h'(1)+f_h(ℓ_h^-)/(1-ℓ_h^-) D_h'(1)   if  D_h'(1)<0,
    ∞   if  D_h'(1)=0,
    
       lim_ξ↑ν_h^+D_h(ϕ_h(ξ))ϕ_h'(ξ) = 0.



  (iii) In all the other cases I_h=ℝ  and

    lim_ξ→±∞ϕ_h'(ξ) = 0.




We observe that for c_h given by (<ref>), we deduce by (f) that g_h(ρ) ≥ 0 for all ρ∈[ℓ_h^-,ℓ^+_h], see  <ref>. Moreover,  we have

    g_h(ℓ_h^+) = g_h(ℓ_h^-) =
    -f_h(ℓ_h^+)  ℓ_h^- - f_h(ℓ_h^-)  ℓ_h^+/ℓ_h^+ - ℓ_h^-

and no ρℓ_h^± makes g_h(ρ) equal to that value.

Theorem <ref> motivates the following definition.


A traveling-wave solution ρ_h is stationary if c_h=0.
It is degenerate if at least one of conditions (i) or (ii) of Theorem <ref> holds.



A consequence of assumption (f) is that if ρ_h is degenerate, then the profile ϕ_h is singular either at ν_h^- in case (i) or at ν_h^+ in case (ii), in the sense that ϕ_h' cannot be extended to the whole of  as a continuous function.


In case (i) (or (ii)) of Theorem <ref> does not hold we define ν_h^- ≐ -∞ (respectively, ν_h^+ ≐∞). In this way the interval (ν_h^-,ν_h^+) is always defined and coincides with the interval I_h defined in (<ref>):

    I_h = (ν_h^-,ν_h^+).

The interval I_h is bounded if and only if both (i) and (ii) hold; in this case ρ_h is both degenerate and stationary.  As a consequence, if ρ_h is non-stationary then I_h is unbounded and coincides either with a half line (if ρ_h is degenerate) or with  (if ρ_h is non-degenerate).
At last, ρ_h is degenerate if and only if either ν_h^- or ν_h^+ is finite.

In the case of non-stationary traveling-wave solutions ρ_h we use the notation

    ω_h ≐min{c_h^-1ν_h^-,c_h^-1ν_h^+}.



Let ρ_h be a traveling-wave solution of (<ref>); then we have the following.


  (a)
If ρ_h is stationary, then it is degenerate if and only if D_h(0)D_h(1)=0 and ℓ_h^- = 0 (hence ℓ_h^+=1).

  (b)
If ρ_h is non-stationary, then it is degenerate if and only if one of the following equivalent statements hold:


  * either D_h(0) = 0 = ℓ_h^- or D_h(1) = 0 = 1-ℓ_h^+, but not both;

  * ω_h is finite.

In this case the function ξ↦ϕ_h'(c_hξ) is singular at ξ = ω_h and 1 elsewhere.



We recall that ρ_h is degenerate if and only if either D_h(0) = 0 = ℓ_h^- or D_h(1) = 0 = 1-ℓ_h^+. This means that at least one of the end states must be 0 or 1, say 0; but then c_h=0 if and only if the other end state is 1. This proves (a) and the first part of (b).

Now, we prove the second part of (b).
Since c_h  0, exactly one between (i) and (ii) of Theorem <ref> occurs, namely, exactly one between ν_h^- and ν_h^+ is finite.
If ν_h^- is finite and ν_h^+=∞, then c_h=f_h(ℓ_h^+)/ℓ_h^+>0 and ω_h = c_h^-1ν_h^- is finite.
By Remark <ref>, we know that ξ↦ϕ_h'(ξ) is singular at ξ=ν_h^- and 1 elsewhere, whence the regularity of ξ↦ϕ_h'(c_hξ).
Analogously, if ν_h^+ is finite and ν_h^-=-∞, then c_h=-f_h(ℓ_h^-)/(1-ℓ_h^-)<0 and ω_h = c_h^-1ν_h^+ is finite. The statement about the smoothness of ξ↦ϕ_h'(c_hξ) is proved as above.

Finally, the converse is straightforward. In fact, if ω_h is finite, then either ω_h = c_h^-1ν_h^- and ν_h^- is finite, or ω_h = c_h^-1ν_h^+ and  ν_h^+ is finite; in both cases ρ_h is degenerate.


Because of the smoothness properties of the profile proved in Theorem <ref>, we can integrate equation (<ref>) in (ξ_-,ξ)⊂ I_h and we obtain

    c_h ϕ_h(ξ) - F_h(ϕ_h(ξ), ϕ_h'(ξ))
    =
    c_h ϕ_h(ξ_-) - F_h(ϕ_h(ξ_-), ϕ_h'(ξ_-)).

If ξ_-↓ν_h^- in the previous expression, by applying (<ref>) or (<ref>) we deduce

    F_h(ϕ_h(ξ), ϕ_h'(ξ)) = c_h ϕ_h(ξ) + g_h(ℓ_h^±),      ξ∈ I_h.

We observe that (<ref>) is trivially satisfied in case (i) when  ξ < ν_h^- and in case (ii) when  ξ > ν_h^+; moreover, by a continuity argument, we deduce from (<ref>) and (<ref>) that (<ref>) is satisfied in case (i) at ξ = ν_h^- and in case (ii) at ξ = ν_h^+, respectively. In conclusion, we have that (<ref>) holds in the whole , namely

    D_h(ϕ_h(ξ))  ϕ_h'(ξ) = g_h(ϕ_h(ξ)) - g_h(ℓ_h^±),      ξ∈.






§ TRAVELING WAVES IN A NETWORK


In this section we consider the traveling-wave solutions of problem (<ref>)-(<ref>) in the network 𝒩.
We first introduce the definition of traveling-wave solution in 𝒩.

For any h ∈𝖧, let ρ_h be a traveling-wave solution of (<ref>)_h in the sense of Definition <ref> and set ρ≐ (ρ_1, …, ρ_m+n).
With reference to  Definition <ref>, we say that:


  * ρ is stationary if each component ρ_h is stationary;

  * ρ is completely non-stationary if none of its components is stationary;

  * ρ is degenerate if at least one component ρ_h is degenerate;

  * ρ is completely degenerate if each of its components is degenerate.

Finally, we say that ρ is a traveling-wave solution of problem (<ref>)-(<ref>) in the network 𝒩 if (<ref>) holds.


For brevity, from now on we simply write traveling wave  for traveling-wave solution. In analogy to the notation above, we say that ϕ≐ (ϕ_1, …, ϕ_m+n) is a profile for ρ if ϕ_h is a profile corresponding to ρ_h for every h ∈𝖧.

For clarity of exposition, we collect our general results for stationary and non-stationary traveling waves in the following subsections.



 §.§ General results


In this subsection, as well as in the following ones, we always assume (f) and (D) without explicitly mentioning it. Moreover, by Definition <ref> and Theorem <ref>, the end states and the speeds of the profiles must satisfy (<ref>) for every h∈𝖧; both conditions in (<ref>) are tacitly assumed as well.


The function ϕ is the profile of a traveling wave if and only if ϕ_h is a solution to (<ref>)-(<ref>) for any h ∈𝖧 and

    c_j  ϕ_j(c_j t) + g_j(ℓ_j^±)
    =
    ∑_i ∈𝖨α_i,j[ c_i  ϕ_i(c_i t) + g_i(ℓ_i^±) ],      t ∈,   j ∈𝖩.

In (<ref>) any combination of the signs ± is allowed.

By plugging ρ_h(t,x)=ϕ_h(x-c_ht) in (<ref>) and recalling that by Theorem <ref> the profiles are continuous in , we obtain

    F_j(ϕ_j(- c_j t), ϕ_j'(- c_j t))
    =
    ∑_i ∈𝖨α_i,j  F_i(ϕ_i(- c_i t), ϕ_i'(- c_i t)),      t∈, j∈𝖩,

which is equivalent to (<ref>) by (<ref>).
At last, we can clearly choose any combination of signs in (<ref>) because of (<ref>).


Differently from what specified in Proposition <ref>, in the following the choice of the signs “±” follows the usual rules, i.e., top with top and bottom with bottom.


Assume that problem (<ref>)-(<ref>) admits a traveling wave. Then for any j∈𝖩 we have

    max{f_j(ℓ_j^-), f_j(ℓ_j^+)}   = ∑_i ∈𝖨α_i,j max{f_i(ℓ_i^-), f_i(ℓ_i^+)},
    
    min{f_j(ℓ_j^-), f_j(ℓ_j^+)}   = ∑_i ∈𝖨α_i,j min{f_i(ℓ_i^-), f_i(ℓ_i^+)}.



Fix j∈𝖩.
We notice that (<ref>) is equivalent to

    Υ_j(t) =
    ∑_i ∈𝖨α_i,jΥ_i(t),   t ∈,   j ∈𝖩,

where the map t ↦Υ_h(t) ≐ c_h  ϕ_h(c_ht) + g_h(ℓ_h^-) is non-decreasing because the profiles are so, by Theorem <ref>. Since we can write Υ_h(t) = f_h(ℓ_h^-) + c_h [ϕ_h(c_ht) - ℓ_h^-], we see that Υ_h ranges between f_h(ℓ_h^-) and f_h(ℓ_h^+) because of (<ref>) and the fact that ξ↦ϕ_h(ξ) takes values in [ℓ_h^-,ℓ_h^+].  As a consequence,

    lim_t→∞Υ_h(t) = max{f_h(ℓ_h^-),f_h(ℓ_h^+)
    },      lim_t→-∞Υ_h(t) = min{f_h(ℓ_h^-),f_h(ℓ_h^+)
    }.

Hence, by passing to the limit for t→±∞ in (<ref>) we obtain (<ref>) and (<ref>), respectively.



Assume that problem (<ref>)-(<ref>) admits a traveling wave. The traveling wave is stationary if and only if one of the following equivalent statements hold:


  (i)
there exists j∈𝖩 such that c_ j = 0;

  (ii)
c_i = 0 for all i ∈𝖨;

  (iii)
c_j = 0 for all j ∈𝖩.



By subtracting (<ref>) to (<ref>) we obtain

    |f_j(ℓ_j^+) - f_j(ℓ_j^-)| = ∑_i ∈𝖨α_i,j |f_i(ℓ_i^+) - f_i(ℓ_i^-)|.

Since c_h=0 if and only if f_h(ℓ_h^-) = f_h(ℓ_h^+), from the above equation we immediately deduce that (i), (ii) and (iii) are equivalent. By the equivalence of (ii) and (iii), a traveling wave is stationary if and only if one of the statements above holds.


Lemma <ref> shows that either a traveling wave is stationary, and then c_h=0 for every h∈𝖧, or it is non-stationary, and then

    there exists i∈𝖨 such that c_ i0 and c_j0 for every j∈𝖩.

Of course, by Lemma <ref>, c_ i0 for some i∈𝖨 if and only if c_j0 for every j∈𝖩.


Fix ℓ_i^±∈ [0,1] with ℓ_i^- < ℓ_i^+, i ∈𝖨.
Then for any j∈𝖩 there exist ℓ_j^±∈ [0,1] with ℓ_j^- < ℓ_j^+ and satisfying (<ref>)-(<ref>) if and only if

    max_[0,1]f_j > ∑_i ∈𝖨α_i,j max{f_i(ℓ_i^-), f_i(ℓ_i^+)}   if c_1=…=c_m=0,
    
    max_[0,1]f_j ≥∑_i ∈𝖨α_i,j max{f_i(ℓ_i^-), f_i(ℓ_i^+)}   otherwise.

In this case, the end states ℓ_j^± are uniquely determined if and only if c_i=0 for every i ∈𝖨.


Assume that there exist ℓ_j^±∈ [0,1], with ℓ_j^- < ℓ_j^+, which satisfy (<ref>)-(<ref>). Then clearly we have max_[0,1]f_j ≥∑_i ∈𝖨α_i,j max{f_i(ℓ_i^-), f_i(ℓ_i^+)}. If c_i=0 for every i∈𝖨, then we have c_j=0 for every j∈𝖩 by Lemma <ref>; the equality max_[0,1]f_j= f(ℓ_j^-)=f(ℓ_j^+) would imply ℓ_j^-=ℓ_j^+ because of (f), a contradiction, and then max_[0,1]f_j> f(ℓ_j^-)=f(ℓ_j^+). This proves (<ref>).



Conversely, assume (<ref>). If c_i=0 for every i∈𝖨, then ℓ_j^-<ℓ_j^+ are uniquely determined because of the strict concavity of f_j. Assume, on the contrary, that c_ i0 for some i∈𝖨; then c_j0 by Lemma <ref>, i.e., f_j(ℓ_j^-) f_j(ℓ_j^+). Thus (<ref>)-(<ref>) determine exactly four possible choices of end states ℓ_j^± with ℓ_j^- < ℓ_j^+, see Figure <ref>.


By Proposition <ref> and Lemma <ref> we deduce that the end states ℓ_j^± are uniquely determined in terms of the end states ℓ_i^± if and only if the traveling wave is stationary and the first condition in (<ref>) holds.

We now give an algebraic result about determining the end states of the outgoing profiles in terms of the end states of the ingoing ones. We introduce

    L_i,j^±≐ℓ_i^±    if  c_i   c_j ≥ 0,
    
    ℓ_i^∓    if  c_i   c_j <0.



Assume that problem (<ref>)-(<ref>) admits a traveling wave. Then for any j∈𝖩 we have

    f_j(ℓ_j^±)    =
    ∑_i ∈𝖨α_i,j  f_i(L_i,j^±).

Moreover, (<ref>) is equivalent to (<ref>)-(<ref>).


Fix j∈𝖩.
By Lemma <ref> it is sufficient to prove that (<ref>) is equivalent to (<ref>)-(<ref>).
If c_j>0, and then f_j(ℓ_j^+) > f_j(ℓ_j^-), by (<ref>) we have

    max{f_i(ℓ_i^-), f_i(ℓ_i^+)}   =
    
    f_i(ℓ_i^+)   if c_i≥0
    
    
    f_i(ℓ_i^-)   if c_i<0
    
    =f_i(L_i,j^+),
    
    min{f_i(ℓ_i^-), f_i(ℓ_i^+)}   =
    
    f_i(ℓ_i^-)   if c_i≥0
    
    
    f_i(ℓ_i^+)   if c_i<0
    
    =f_i(L_i,j^-),

and therefore (<ref>) is equivalent to (<ref>)-(<ref>).
The case c_j<0 is analogous.
If c_j=0, then f_j(ℓ_j^+) = f_j(ℓ_j^-) and by Lemma <ref> we have f_i(ℓ_i^+) = f_i(ℓ_i^-) for any i ∈𝖨. In this case formulas (<ref>)-(<ref>) reduce to a single equation, which coincides with (<ref>).




 §.§ The stationary case


In this short subsection we briefly consider stationary traveling waves.


Problem (<ref>)-(<ref>) admits infinitely many stationary traveling waves; such waves are characterized by the conditions on the end states

    f_h(ℓ_h^+) = f_h(ℓ_h^-),      f_j(ℓ_j^-) = ∑_i ∈𝖨α_i,j  f_i(ℓ_i^-)    for h∈𝖧, j∈𝖩.



Clearly, (<ref>) is trivially satisfied if ℓ_h^-=0 and ℓ_h^+ = 1 for all h ∈𝖧. We claim that there exist infinitely many choices of ℓ_1^±,…,ℓ_m+n^± satisfying (<ref>).  To prove the claim, we choose ℓ_i^±∈ [0,1], with ℓ_i^- < ℓ_i^+, such that f_i(ℓ_i^-) = f_i(ℓ_i^+) are sufficiently small to satisfy the first condition in (<ref>) for all j ∈𝖩.
Then, by a continuity argument, we can choose ℓ_j^±∈ [0,1] so that ℓ_j^- < ℓ_j^+ and f_j(ℓ_j^-) = f_j(ℓ_j^+) = ∑_i ∈𝖨α_i,j  f_i(ℓ_i^-). This proves the claim.

With this choice of the end states, by Theorem <ref> we deduce the existence of a stationary traveling wave in each road satisfying (<ref>). At last we notice that, in the stationary case, condition (<ref>) is equivalent to the latter condition in (<ref>).


Clearly, if both D_h(0)0 and D_h(1)0 for every h∈𝖧, then problem (<ref>)-(<ref>) admits no degenerate traveling wave. However, even in the general case, the proof of Theorem <ref> shows that (<ref>)-(<ref>) admits infinitely many non-degenerate stationary traveling waves: just choose 0ℓ_h^-<ℓ_h^+1 satisfying (<ref>). Moreover, if there exists h∈𝖧 such that either D_ h(0)=0 or D_ h(1)=0, then (<ref>)-(<ref>) admits also infinitely many degenerate stationary traveling waves: just choose ℓ_ h^-=0 = 1 - ℓ_ h^+ and determine the other end states by (<ref>).



 §.§ The non-stationary case


In this subsection we consider non-stationary traveling waves.
By Lemma <ref> this is equivalent to consider the scenario in (<ref>): there exists i∈𝖨 such that f_ i(ℓ_ i^-)  f_ i(ℓ_ i^+) and f_j(ℓ_j^-)  f_j(ℓ_j^+) for every j∈𝖩.
We can therefore introduce the following notation:

    c_i,j≐c_i/c_j,       A_i,j≐α_i,j  c_i,j,
       k_j ≐∑_i ∈[ A_i,j  L_i,j^±] - ℓ_j^±,      κ_j
    ≐ c_jk_j,

where L_i,j is defined in (<ref>) and

    ≐{i∈𝖨:c_i=0} = {i∈𝖨:f_i(ℓ_i^-) = f_i(ℓ_i^+)},      ≐𝖨∖.

We notice that ∅ by (<ref>) and that both  and  depend on the end states ℓ_i^±, i∈𝖨, indeed.
Moreover, k_j is well defined because by (<ref>)

    ∑_i ∈ A_i,j[ L_i,j^+ - L_i,j^- ] =
    ∑_i ∈α_i,j c_j^-1[ f_i(L_i,j^+) - f_i(L_i,j^-) ] =
    c_j^-1[ f_j(ℓ_j^+) - f_i(ℓ_j^-) ] = ℓ_j^+-ℓ_j^-.

Finally, by (f) we deduce that

    for no j ∈𝖩 we have both ℓ_j^-=0=1-ℓ_j^+.



The function ϕ is the profile of a non-stationary traveling wave if and only if ϕ_h is a solution to (<ref>)-(<ref>) for any h ∈𝖧 and

    ϕ_j(ξ)     =
    ∑_i ∈[ A_i,j ϕ_i(c_i,j ξ) ]
    - k_j,      ξ∈, j∈𝖩.



By Proposition <ref> it is sufficient to prove that by (<ref>) condition (<ref>) is equivalent to (<ref>).
By (<ref>) we have g_i(ℓ_i^+) = g_i(ℓ_i^-) = g_i(L_i,j^+) = g_i(L_i,j^-) and then by (<ref>) we have κ_j = g_j(ℓ_j^±) - ∑_i ∈𝖨α_i,j g_i(L_i,j^±).
Hence, by (<ref>), with the change of variable ξ = c_jt, condition (<ref>) is

    c_j  ϕ_j(ξ) =
    - g_j(ℓ_j^±) + ∑_i ∈𝖨α_i,j [ c_i  ϕ_i(c_i,jξ) + g_i(L_i,j^±) ]
    =
    ∑_i ∈𝖨α_i,j  c_i  ϕ_i(c_i,jξ) - κ_j,

that is equivalent to (<ref>).


We observe that k_j and (<ref>) can be written in a little bit more explicit form by avoiding the use of L_i,j^± as follows

    k_j    = ∑_i ∈[ A_i,j ℓ_i^- + ℓ_i^+/2] - ℓ_j^- + ℓ_j^+/2,
    
    ϕ_j(ξ)
       =
    ℓ_j^- + ℓ_j^+/2
    +
    ∑_i ∈ A_i,j[ ϕ_i(c_i,j ξ) - ℓ_i^- + ℓ_i^+/2].

Proposition <ref> shows how each outgoing profile ϕ_j can be expressed by (<ref>) in terms of the ingoing profiles ϕ_i, i∈𝖨.
We know a priori that ϕ_j is increasing and its end states are contained in the interval [0,1].
Now, we prove a sort of converse implication, which shows that these properties of the profile ϕ_j are enjoined by the function defined by the right-hand side of (<ref>).


Let ϕ_i,  for i∈𝖨, be the profiles provided by Theorem <ref> and assume that ∅; fix j∈𝖩 and consider any l_j^±∈[0,1] satisfying (<ref>) and such that, for the corresponding c_j, it holds c_j0.
Then l_j^- < l_j^+. Moreover, denote by ℓ_j(ξ) the right-hand side of (<ref>); then ξ↦ℓ_j(ξ) is non-decreasing and ℓ_j(±∞)=l_j^±.



Since by Theorem <ref> we know that ℓ_i^-<ℓ_i^+, then by (<ref>)

    l_j^+-l_j^- =
    c_j^-1[ f_j(l_j^+) - f_j(l_j^-) ] =
    ∑_i ∈𝖨α_i,j  c_j^-1[ f_i(L_i,j^+) - f_i(L_i,j^-) ]
    
    =
    ∑_i ∈𝖨α_i,j  c_i,j[L_i,j^+- L_i,j^-] =
    ∑_i ∈α_i,j  |c_i,j| [ℓ_i^+ - ℓ_i^-]    > 0.

By definition of ℓ_j we have
ℓ_j'(ξ) =
∑_i ∈α_i,j  c_i,j^2  ϕ_i'(c_i,j ξ)
for a.e. ξ∈, hence ξ↦ℓ_j(ξ) is non-decreasing since all profiles ϕ_i do.
Moreover, ℓ_j(±∞) = l_j^± because by the definitions of ℓ_j and κ_j  we have

    c_j ℓ_j(±∞) =
    ∑_i ∈[ α_i,j  c_i   L_i,j^±]
    - κ_j =
    c_j   l_j^±.



We notice that Proposition <ref> exploits condition (<ref>) through its expression (<ref>) for the profiles; the diffusivities D_h are not involved in (<ref>). Indeed, Proposition <ref> imposes strong necessary conditions on the diffusivities as we discuss now as a preparation to (<ref>).

We notice that if both ν_h^- and ν_h^+ are finite, then ℓ_h^-=0=1-ℓ_h^+ and consequently c_h=0; therefore either ν_h^- or ν_h^+ (possibly both) is infinite for any h∈∪𝖩.

The following result is similar to Lemma <ref>.

Problem (<ref>)-(<ref>) admits a degenerate non-stationary traveling wave ρ if and only if at least one of the following conditions holds:


  (A) for some i ∈ we have D_i(0) D_i(1)= 0 and ℓ_i^- = 0 (hence ℓ_i^+=1);

  (B) for every h ∈∪𝖩 we have either D_h(0) = 0 = ℓ_h^- or D_h(1) = 0 = ℓ_h^+-1, but not both. In this case we have

    ω_i = ω_j≐ω,      i∈, j ∈𝖩.




Let us introduce the following conditions:


  (B)' for some i ∈ we have either D_i(0) = 0 = ℓ_i^- or D_i(1) = 0 = ℓ_i^+-1, but not both;

  (B)” for some j ∈𝖩 we have either D_j(0) = 0 = ℓ_j^- or D_j(1) = 0 = ℓ_j^+-1, but not both.

Clearly (B) implies both (B)' and (B)”.
Moreover, by Lemma <ref> and (<ref>), problem (<ref>)-(<ref>) admits a degenerate non-stationary traveling wave ρ if and only if at least one of the conditions (A), (B)' and (B)” holds.
To complete the proof it is therefore sufficient to show that (B), (B)' and (B)” are equivalent.
By Lemma <ref>(b) and (<ref>), the conditions (B), (B)' and (B)” are respectively equivalent to


  (I) ω_h is finite for every h ∈∪𝖩,

  (II) for some i∈ we have that ω_i is finite,

  (III) for some j∈𝖩 we have that ω_j is finite,

where ω_h is defined in (<ref>).
Differentiating (<ref>) gives

    ϕ_j'(c_jξ)     =
    ∑_i ∈α_i,j c_i,j^2 ϕ_i'(c_iξ)       for a.e. ξ∈, j ∈𝖩.

More precisely, by Lemma <ref>, formula (<ref>) holds for ξ∈∖({ω_j}∪⋃_i∈ω_i); moreover, by the same lemma we know that ξ↦ϕ_h'(c_hξ) is singular at ξ = ω_h and 1 elsewhere, for h∈∪𝖩.
Hence, (<ref>) implies (<ref>). By (<ref>) we have that the above statements (I), (II) and (III) are equivalent and then also (B), (B)' and (B)” are so.


As for Lemma <ref>, we notice that Lemma <ref> implies that a non-stationary traveling wave ρ is either non-degenerate, and then ρ_h is non-degenerate for every h ∈𝖧, or ρ is degenerate, and then either there exists i∈ such that ρ_ i is degenerate, or ρ_h is degenerate for all h ∈∪𝖩.
In both cases a non-stationary traveling wave ρ satisfies (<ref>).

When modeling traffic flows it is natural to use different diffusivities, which however share some common properties.
For instance, this led to consider in <cit.> the following subcase of (D):



  (D1) D_h satisfies (D) and D_h(0)=0, D_h(1)>0, for every h∈𝖧.



The proof of the following result is an immediate consequence of Lemma <ref> and, hence, omitted.


Assume that problem (<ref>)-(<ref>) has a non-stationary traveling wave ρ and  (D1) holds. Then ρ is degenerate if and only at least one of the following conditions holds:


  (A) for some i ∈ we have ℓ_i^- = 0 (hence ℓ_i^+ = 1);

  (B) for every h ∈∪𝖩 we have ℓ_h^- = 0 (hence ℓ_h^+ ≠ 1).



The case when D_h satisfies (D) and D_h(0)=0=D_h(1) for every h∈𝖧, see <cit.>, can be dealt analogously.

The next result is the most important of this paper; there, we give necessary and sufficient conditions for the existence of non-stationary traveling waves in a network. About its statement, let us recall Theorem <ref>: we have ϕ_h'(ξ)=0 in case (i) if ξ<ν_h^- or in case (ii) if ξ>ν_h^+. Since ϕ_h satisfies equation (<ref>), we are led to extend the quotient ℓ↦g_h(ℓ)-g_h(ℓ_h^-)/D_h(ℓ) to the whole of  by defining

    γ_h(ℓ) ≐g_h(ℓ)-g_h(ℓ_h^-)/D_h(ℓ)    if D_h(ℓ)0,
    
    
    0    if D_h(ℓ)=0.

In fact, when ℓ is replaced by ϕ_h(ξ), then γ_h(ℓ)=ϕ_h'(ξ) for ξ∈∖{ν_h^-, ν_h^+}. We remark that condition D_h(ℓ)=0 occurs at most when either ℓ=0 or ℓ=1. To avoid the introduction of the new notation (<ref>), in the following we simply keep on writing g_h(ℓ)-g_h(ℓ_h^-)/D_h(ℓ) for γ_h(ℓ). As a consequence, any non-stationary traveling wave of problem (<ref>)-(<ref>) satisfies

    ϕ_h'(ξ)
    =
    g_h(ϕ_h(ξ)) - g_h(ℓ_h^-)D_h(ϕ_h(ξ)),      ξ∈∖{ν_h^-,  ν_h^+},  h ∈𝖧.



Assume conditions (f) and (D).
Problem (<ref>)-(<ref>) admits a non-stationary traveling wave if and only if the following condition holds.



  (𝒯) There exist ℓ_1^±,…,ℓ_m^±∈ [0,1] with ℓ_i^- < ℓ_i^+, i∈𝖨, such that:



  (i)  ∅;


  (ii) for any j ∈𝖩 there exist ℓ_j^±∈[0,1] satisfying (<ref>) and such that f_j(ℓ_j^-)  f_j(ℓ_j^+);


  (iii) for any j ∈𝖩 we have

    g_j(ℓ_j(c_j  ξ)) - g_j(ℓ_j^-)D_j(ℓ_j(c_j  ξ))
    =
    ∑_i ∈ A_i,j  c_i,j g_i(ϕ_i( c_i  ξ)) - g_i(ℓ_i^-)D_i(ϕ_i( c_i  ξ))      for a.e. ξ∈,

where ϕ_1,…,ϕ_m are solutions to (<ref>)-(<ref>) and, for k_j as in (<ref>),

    ℓ_j(ξ) ≐∑_i ∈[ A_i,j ϕ_i(c_i,j ξ) ]
    - k_j,      ξ∈.






First, assume that problem (<ref>)-(<ref>) admits a non-stationary traveling wave ρ with profiles ϕ_h, end states ℓ_h^± and speeds c_h, for h∈𝖧. By Theorem <ref> we have that ℓ_h^± and c_h satisfy (<ref>).
By Proposition <ref> the profiles ϕ_h satisfy (<ref>)-(<ref>) and (<ref>).
The end states ℓ_j^±, j∈𝖩, satisfy (<ref>) by Proposition <ref>.
Since ρ is non-stationary we are in the scenario given by (<ref>): ∅ and f_j(ℓ_j^-)  f_j(ℓ_j^+) for all j ∈𝖩.
By (<ref>) with h = j we have

    ϕ_j'(c_jξ)
    =
    g_j(ϕ_j(c_jξ)) - g_j(ℓ_j^-)D_j(ϕ_j(c_jξ))

for ξ∈ in the non-degenerate case and for ξ∈∖{ω} with ω given by (<ref>) in the degenerate case.
On the other hand, by differentiating (<ref>) and applying (<ref>) with h = i we deduce

    ϕ_j'(ξ)    =
    ∑_i ∈ A_i,j  c_i,j ϕ_i'( c_i,jξ)
    =
    ∑_i ∈ A_i,j  c_i,j g_i(ϕ_i( c_i,jξ)) - g_i(ℓ_i^-)D_i(ϕ_i( c_i,jξ))

for ξ∈ in the non-degenerate case and for ξ∈∖{ω} with ω given by (<ref>) in the degenerate case.
Identity (<ref>) follows because ℓ_j ≡ϕ_j by (<ref>) and by comparing (<ref>), (<ref>).

Conversely, assume that condition (𝒯) holds.
We remark that the existence of ϕ_i, i ∈𝖨, is assured by Theorem <ref>.
Fix  j∈𝖩.
By defining ϕ_j ≐ℓ_j we obtain (<ref>).
We know by assumption that ∅, ℓ_j^±∈ [0,1] satisfy (<ref>) and c_j  0; we can apply therefore Lemma <ref> and deduce that ℓ_j^-<ℓ_j^+, ϕ_j is non-decreasing and satisfies (<ref>) with h=j.
By Proposition <ref>, what remains to prove is that ϕ_j satisfies (<ref>).
But by (<ref>) we deduce (<ref>) for a.e. ξ∈, because ϕ_1,…,ϕ_m satisfy (<ref>) and hence, recalling the extension (<ref>), also (<ref>); then by (<ref>) we conclude that ϕ_j satisfies (<ref>) for a.e. ξ∈ and then  (<ref>) for a.e. ξ∈.
Finally, (<ref>) holds by the regularity ensured by Theorem <ref> for the profiles.



Fix ℓ_i^±∈ [0,1], i ∈𝖨, so that ℓ_i^- < ℓ_i^+ and (<ref>) holds. We know by Proposition <ref> that for every j∈𝖩 there exists (ℓ_j^-,ℓ_j^+) ∈ [0,1]^2, with ℓ_j^- < ℓ_j^+, that satisfies (<ref>), but it is not unique. If beside (<ref>) we impose also (<ref>), then we may have three possible scenarios: such (ℓ_j^-,ℓ_j^+) either does not exist, or it exists and is unique, or else it exists but is not unique. We refer to Subsections <ref> and <ref> for further discussion.






§ THE CONTINUITY CONDITION


In this section we discuss the case when solutions to (<ref>)-(<ref>) are also required to satisfy the continuity condition (<ref>); this makes the analysis much easier because (<ref>) implies several strong conditions.

First, we provide the main results about traveling waves satisfying condition (<ref>). We point out that some of the consequences below have already been pointed out in <cit.> in the case that some Kirchhoff conditions replace the conservation of the total flow (<ref>). In order to emphasize the consequences of the continuity condition (<ref>), the first two parts of the following lemma do not assume that also condition (<ref>) holds.

For any h ∈𝖧, let ρ_h be a traveling wave of (<ref>)_h in the sense of Definition <ref> and set ρ≐ (ρ_1, …, ρ_m+n); then the following holds for every (i,j) ∈𝖨×𝖩 and h∈𝖧.


  (i) ρ satisfies (<ref>) if and only if

    ϕ_j(c_jt) = ϕ_i(c_i t) ≐Φ(t),      t∈.


  (ii) If ρ satisfies (<ref>), then either it is stationary (hence (<ref>) reduces to ϕ_j(0) = ϕ_i(0)), or it is completely non-stationary and the speeds c_h have the same sign (hence c_i,j>0).
In the latter case, ρ is either non-degenerate or completely degenerate; moreover

    (c_j^-1I_j)    = (c_i^-1I_i) ≐ℐ,
    
    ℓ_j^±   =ℓ_i^±=L_i,j^±≐ℓ^±,
    
    
    c_jg_j(ℓ)-g_j(ℓ^±)/D_j(ℓ)   = c_ig_i(ℓ)-g_i(ℓ^±)/D_i(ℓ),      ℓ∈(ℓ^-,ℓ^+).


  (iii)
If ρ is non-stationary and satisfies both (<ref>) and (<ref>), then

    c_j =∑_i∈𝖨α_i,jc_i,      ∑_j∈𝖩c_j =∑_i∈𝖨c_i,      κ_j=0,      ∑_i ∈𝖨 A_i,j = 1.





We split the proof according to the items in the statement.






  * Condition (<ref>) and (<ref>) are clearly equivalent.


  * Since we are discarding constant profiles, by (<ref>) we have that either c_h=0 for all h∈𝖧 or c_h0 for all h∈𝖧. The stationary case is trivial; therefore we consider below only the non-stationary case and assume that c_h0 for all h∈𝖧.
By differentiating (<ref>) with respect to t we deduce

    c_jϕ_j'(c_jt) = c_iϕ_i'(c_i t)       for a.e. t∈.

Then (<ref>) implies that either ρ is non-degenerate or it is completely degenerate.
Moreover (<ref>) implies (<ref>) because, by Lemma <ref>, we have that ρ_h is degenerate if and only if the map ξ↦ϕ_h'(c_h ξ) is singular at ξ = ω_h ∈ and 1 elsewhere.
By taking t ∈ℐ in (<ref>) we deduce that c_i and c_j have the same sign. As a consequence we have L_i,j^±=ℓ_i^± and then ℓ_i^± = ℓ_j^± by letting t→±∞ in (<ref>).
By (<ref>), (<ref>) and (<ref>) we have D_h(Φ(ξ)) Φ'(ξ) = c_h [g_h(Φ(ξ)) - g_h(ℓ^±)] for all h∈𝖧, whence (<ref>) by the extension (<ref>).


  * To deduce (<ref>)_1, we differentiate (<ref>) and then exploit (<ref>). Formula (<ref>)_2 follows by summing (<ref>)_1 with respect to j and by (<ref>).
By (<ref>)_1 we have ∑_i ∈𝖨 A_i,j = ∑_i ∈𝖨α_i,j c_i c_j^-1=1, which proves (<ref>)_4.
Finally, (<ref>) and (<ref>)_4 imply (<ref>)_3.


In the following proposition we deal with stationary traveling waves satisfying condition (<ref>).


Problem (<ref>)-(<ref>) admits infinitely many stationary traveling waves satisfying (<ref>); their end states ℓ_h^± satisfy (<ref>) and are such that S ≐⋂_h∈𝖧(ℓ_h^-,ℓ_h^+)∅.


By (<ref>) condition (<ref>) holds in the stationary case if and only if ϕ_i(0)=ϕ_j(0) for (i,j)∈𝖨×𝖩.
Recalling the proof of Theorem <ref>, it is sufficient to take ℓ_h^±∈ [0,1] satisfying (<ref>) and such that S ∅, ℓ_0 ∈ S and the unique solution ϕ_h to (<ref>)-(<ref>) such that ϕ_h(0) = ℓ_0. There are infinitely many of such profiles because of the arbitrariness of ℓ_h^±.


We point out that condition S=∅ can occur if the functions f_h assume their maximum values at different points. This is not the case when the following condition (<ref>)_1 is assumed.

The following result is analogous to Theorem <ref> in the case (<ref>) holds.


Assume conditions (f) and (D). Problem (<ref>)-(<ref>) admits a (completely) non-stationary  traveling wave satisfying (<ref>) if and only if the following condition holds.


  (𝒯_c)  There exist ℓ^±∈ [0,1] with ℓ^- < ℓ^+, such that for any h∈𝖧, i ∈𝖨 and j ∈𝖩

    f_h(ℓ^-)     f_h(ℓ^+),
    
    
    f_j(ℓ^±)   =∑_i ∈𝖨α_i,jf_i(ℓ^±),
    
    
    c_j g_j(ϕ_j(c_j t)) - g_j(ℓ^-)D_j(ϕ_j(c_j t))   =
    c_i g_i(ϕ_i(c_i t)) - g_i(ℓ^-)D_i(ϕ_i(c_i t))      for a.e. t∈,

where c_h is given by (<ref>), ϕ_h is a solution to (<ref>) such that φ_h(±∞) = ℓ^± and φ_1(0) = … = φ_m+n(0).



Assume that condition (𝒯_c) holds; the other implication is obvious.
We remark that the existence of ϕ_1,…,ϕ_m+n is assured by Theorem <ref>; indeed, for any ℓ_0 ∈ (ℓ^-,ℓ^+), up to shifts it is always possible to assume that φ_h(0) = ℓ_0, h ∈𝖧.
By (<ref>) we have (<ref>)_4 because

    ∑_i ∈𝖨 A_i,j = ∑_i ∈𝖨α_i,jf_i(ℓ^+) - f_i(ℓ^-)/f_j(ℓ^+) - f_j(ℓ^-) = 1.

By (<ref>) we have that = 𝖨, = ∅ and ρ corresponding to the profile φ≐ (φ_1, …, φ_m+n) is completely non-stationary.
Then (<ref>)_4 and (<ref>) imply (<ref>)_3, namely k_j = 0.
By Lemma <ref> (i) and Proposition <ref> it remains to prove (<ref>) and (<ref>).
We start with (<ref>).
Clearly (<ref>) holds for t=0 because ϕ_h(0) = ℓ_0, h∈𝖧.
Then by the extension (<ref>) and (<ref>) we have

    d/ d t[ ϕ_j(c_jt) - ϕ_i(c_i t) ]
    =
    c_j g_j(ϕ_j(c_j t)) - g_j(ℓ^-)D_j(ϕ_j(c_j t))
    -
    c_i g_i(ϕ_i(c_i t)) - g_i(ℓ^-)D_i(ϕ_i(c_i t))
    = 0.

Therefore we conclude that (<ref>) holds.
Finally, (<ref>) follows immediately from (<ref>), (<ref>)_3 and (<ref>)_4.


Consider in particular the case when the functions f and D satisfy (f) and (D), respectively, and assume that

    f_h(ℓ) ≐ v_h f(ℓ),      D_h(ℓ) ≐δ_h D(ℓ),      ℓ∈ [0,1],

for some constants v_h,δ_h>0. Denote

    v_i,j≐v_i/v_j,      δ_i,j≐δ_i/δ_j.

We notice that now we have

    v_i,j=c_i,j.

In the following proposition we apply Theorem <ref> when (<ref>) is assumed; in this case conditions (<ref>) and (<ref>) no longer depend on the end states and the statement is somewhat simplified.


Assume (<ref>) with f and D satisfying (f) and (D), respectively. Problem (<ref>)-(<ref>) admits a (completely) non-stationary traveling wave satisfying (<ref>) if and only if for every i∈𝖨 and j∈𝖩 we have

    v_i,j^2=δ_i,j    and       ∑_i ∈𝖨α_i,j v_i,j = 1.



We only need to translate condition (𝒯_c) to the current case. Let ℓ^±∈ [0,1] with ℓ^-<ℓ^+ and f(ℓ^-) f(ℓ^+).
By (<ref>) it is obvious that (<ref>) is satisfied.
If ℓ^-=0 or ℓ^+=1 condition (<ref>) is satisfied by (f). In all the other cases (<ref>) is equivalent to
∑_i ∈𝖨α_i,j v_i,j = 1
by (<ref>).
Similarly, condition (<ref>) reduces to c_jv_jδ_j^-1=c_iv_iδ_i^-1 and hence, by (<ref>), it is equivalent to v_i,j^2=δ_i,j.



Remark that by (<ref>) condition (<ref>)_2 is equivalent to (<ref>)_4.





§ APPLICATION TO THE CASE OF A QUADRATIC FLUX, 


In this section we assume (<ref>) for some constants v_h,δ_h>0,  D satisfying (D) and the quadratic flux <cit.>

    f(ρ) ≐ρ  (1-ρ),

with no further mention. 
The case when only (<ref>)_1 holds is doable and follows with slight modifications. We use the notation introduced in (<ref>).

For simplicity, in the whole section we focus on the case m=1, see Figure <ref>, even without explicitly mentioning it. Then 𝖨={1}, 𝖩={2,…,n+1}, 𝖧={1,2,…, n+1}. The general case m>1 offers no further difficulties than heavier calculations. 







In this case, condition (<ref>) becomes

    0≤ℓ_h^- < ℓ_h^+≤1       and       c_h = v_h   [1-ℓ_h^+ - ℓ_h^-].

In particular, by (<ref>)_2

    ρ_h is stationary ℓ_h^++ℓ_h^-=1.

Moreover, g_h(ℓ) = v_h  ℓ  [ℓ_h^+ + ℓ_h^ - - ℓ] implies

    g_h(ℓ) - g_h(ℓ_h^±) = v_h   (ℓ_h^+ - ℓ)   (ℓ - ℓ_h^-),

and therefore (<ref>) becomes

    δ_h   D(ϕ_h(ξ))  ϕ_h'(ξ) =
    v_h [ℓ_h^+-ϕ_h(ξ)] [ϕ_h(ξ)-ℓ_h^-],      ξ∈.


We first consider stationary traveling waves and specify Theorem <ref> and Proposition <ref> in the current framework.
We define the intervals

    ℒ_j^0≐[0,1/2)   if α_1,j  v_1,j≤ 1,
    
    [0, 1 - √(1-α_1,j^-1  v_1,j^-1)/2)   if α_1,j  v_1,j > 1,
          j ∈𝖩.



Problem (<ref>)-(<ref>) admits infinitely many stationary traveling waves; their end states are characterized by the conditions

    ℓ_1^- ∈⋂_j ∈𝖩ℒ_j^0,      ℓ_1^+ + ℓ_1^- = 1,      ℓ_j^± = 1/2[
    1±√(1-4  α_1,j  v_1,j ℓ_1^+  ℓ_1^-)],      j ∈𝖩.

Moreover, up to shifts, any stationary traveling wave satisfies (<ref>).



The first part of the proposition follows from Theorem <ref>.
Indeed, conditions (<ref>), (<ref>)_1 and (<ref>) are satisfied if and only if for any h∈𝖧 and j∈𝖩

    ℓ_h^- ∈ [0, 1/2),      ℓ_h^+ + ℓ_h^- = 1,      ℓ_j^-   (1 - ℓ_j^-) = α_1,j  v_1,j ℓ_1^-   (1 - ℓ_1^-);

then it is sufficient to compute ℓ_j^± and to observe that the definition of ℒ_j^0 guarantees that they are real numbers. 

The latter part of the proposition is deduced by Proposition <ref> because 1/2 ∈S≐⋂_h ∈𝖧(ℓ_h^-, ℓ_h^+) ∅. 


In the following we treat the existence of non-stationary traveling waves. Since m=1, by Lemma <ref> this is equivalent to assume c_h 0 for h ∈𝖧, namely, the traveling wave is completely non-stationary. 
By (<ref>), (<ref>) and (<ref>), from (<ref>) we deduce

    c_1,j   =v_1,j1-ℓ_1^+-ℓ_1^-/1-ℓ_j^+-ℓ_j^-,    A_1,j   =α_1,jv_1,j1-ℓ_1^+-ℓ_1^-/1-ℓ_j^+-ℓ_j^-,
    
    
    k_j    =
    A_1,j  L_1,j^± - ℓ_j^±,   κ_j   =v_jℓ_j^-ℓ_j^+-α_1,jv_1ℓ_1^-ℓ_1^+.


The following result translates Theorem <ref> to the present case.
We define the intervals

    ℒ_j^c≐
    [0,1]   if α_1,j  v_1,j≤ 1,
    
    
    [0,1] ∖(1 - √(1-α_1,j^-1  v_1,j^-1)/2 , 1 + √(1-α_1,j^-1  v_1,j^-1)/2)    if α_1,j  v_1,j > 1,
          j ∈𝖩.



Problem (<ref>)-(<ref>) admits a (completely) non-stationary traveling wave if and only if the following condition holds.


  (𝒯_q)  There exist ℓ_1^±∈ [0,1] with ℓ_1^-<ℓ_1^+ such that:



  (i)  ℓ_1^++ℓ_1^- 1;


  (ii) ℓ_1^±∈⋂_j ∈𝖩ℒ_j^c;


  (iii) for any j ∈𝖩 we have

    D(ℓ)=  α_1,j δ_1,j/v_1,j  D(ℓ+k_j/A_1,j),      ℓ∈(ℓ_j^-,ℓ_j^+),

where k_j is defined in (<ref>) with ℓ_j^± being solutions to

    ℓ_j^±(1-ℓ_j^±)=α_1,jv_1,jL_1,j^±(1-L_1,j^±).






The proof consists in showing that, in the present case, condition (𝒯) of Theorem <ref> is equivalent to (𝒯_q).


∙ The first item of (𝒯) is clearly equivalent to the first item of (𝒯_q).


∙ We prove now that the second item of (𝒯) is equivalent to the second item of (𝒯_q).

“⇒”
Assume that for any j ∈𝖩 there exist ℓ_j^±∈[0,1] satisfying (<ref>) and such that f_j(ℓ_j^-)  f_j(ℓ_j^+).
Fix j ∈𝖩.
Clearly (<ref>) is equivalent to (<ref>).

If we denote z_1,j^±≐ 4α_1,jv_1,jL_1,j^±(1-L_1,j^±), then the ℓ_j^±-solutions to (<ref>) are, see  <ref>,

    ℓ_j^- = 1/2[1-√(1-z_1,j^-)],
    
    ℓ_j^+ ∈{1/2[1±√(1-z_1,j^+)]
    },
     if c_j>0,      ℓ_j^- ∈{1/2[1±√(1-z_1,j^-)]
    },
    
    ℓ_j^+ = 1/2[1+√(1-z_1,j^+)],
     if c_j<0.

The square roots in (<ref>) are real numbers if and only if z_1,j^±≤1, namely,

    ℓ_1^±(1-ℓ_1^±) ≤ (4α_1,jv_1,j)^-1.

It is easy to see that the above estimate is equivalent to require ℓ_1^±∈ℒ_j^c.

“⇐”
Assume that ℓ_1^±∈⋂_j ∈𝖩ℒ_j^c and fix j ∈𝖩.
The square roots in (<ref>) are then real numbers and ℓ_j^± given in (<ref>) satisfy (<ref>), namely (<ref>).
Obviously ℓ_j^± belong to [0,1].
Finally, since ℓ_j^± are solutions to (<ref>), it is easy to see that f_j(ℓ_j^+)  f_j(ℓ_j^-) because f_1(ℓ_1^+)  f_1(ℓ_1^-).


∙ We prove now that (𝒯) implies the last item of (𝒯_q).
Since the first two items in (𝒯) are equivalent to the first two items in (𝒯_q), we can assume that ℓ_1^++ℓ_1^- 1, ℓ_1^±∈⋂_j ∈𝖩ℒ_j^c and that for any j ∈𝖩 we have (<ref>), namely,

    (ℓ_j^+-ℓ_j(c_j  ξ)) (ℓ_j(c_j  ξ)-ℓ_j^-)D(ℓ_j(c_j  ξ))
    =
    A_1,j  c_1,j  v_1,j/δ_1,j (ℓ_1^+-ϕ_1( c_1  ξ))  (ϕ_1( c_1  ξ)-ℓ_1^-)D(ϕ_1( c_1  ξ))

for a.e. ξ∈, where ϕ_1 is a solution to (<ref>)-(<ref>) and

    ℓ_j(ξ) ≐ A_1,j[ ϕ_1(c_1,j ξ)-L_1,j^±]+ ℓ_j^±,      ξ∈.

We point out that the above expression of ℓ_j is deduced from (<ref>) by applying (<ref>); moreover (<ref>) is deduced from (<ref>) by applying (<ref>).
Recall that both fractions in (<ref>) are meant as in (<ref>).
Since

    (ℓ_j^+-ℓ_j(c_j  ξ)) (ℓ_j(c_j  ξ)-ℓ_j^-)    =
    A_1,j^2 ( L_1,j^+ - ϕ_1(c_1 ξ) ) ( ϕ_1(c_1 ξ) - L_1,j^- ),
    
    (ℓ_1^+-ϕ_1( c_1  ξ))  (ϕ_1( c_1  ξ)-ℓ_1^-)    =
    (L_1,j^+-ϕ_1( c_1  ξ))  (ϕ_1( c_1  ξ)-L_1,j^-),

we have that (<ref>) is equivalent to

    D(ℓ_j(c_jξ)) = α_1,jδ_1,jv_1,j D(ϕ_1(c_1  ξ))      for a.e. ξ∈.

To conclude now that the above condition is equivalent to (<ref>) it is sufficient to recall that by Lemma <ref> the continuous function ξ↦ℓ_j(ξ) is increasing and ℓ_j(±∞) = ℓ_j^± and that ℓ_j(ξ) = A_1,j ϕ_1(c_1,j ξ) - k_j by (<ref>).


∙ Finally, to prove that (𝒯_q) implies the last item of (𝒯) it is enough to trace backwards the proof of the previous item.


We notice that if D is a polynomial with degree 𝔡, then (<ref>) is equivalent to 𝔡 + 1 conditions on the parameters, see for instance (<ref>) and (<ref>).


We point out that by Proposition <ref> we have that problem (<ref>)-(<ref>) admits a (completely) non-stationary traveling wave satisfying (<ref>) if and only if

    v_1,j^2=δ_1,j    and       α_1,j v_1,j = 1,      j∈𝖩.



The special cases of constant or linear diffusivities are treated in the following subsections.



 §.§ The case of constant diffusivities


In this subsection we assume

    D ≐ 1,

and in this case problem (<ref>)-(<ref>) reduces to

    δ_h  ϕ_h'(ξ) = v_h [ℓ_h^+-ϕ_h(ξ)][ϕ_h(ξ)-ℓ_h^-],   ξ∈,
    
    ϕ_h(±∞) = ℓ_h^±.

For any h∈𝖧, the function

    ψ_h(ξ)
    ≐ℓ_h^+1+e^-v_h/δ_h[ℓ_h^+-ℓ_h^-] ξ
    +
    ℓ_h^-1+e^v_h/δ_h[ℓ_h^+-ℓ_h^-] ξ

solves (<ref>) because ℓ_h^- <ℓ_h^+; all the other solutions are of the form ϕ_h(ξ)=ψ_h(ξ+σ_h) for σ_h ∈. Notice that ψ_h(0)=(ℓ_h^+ + ℓ_h^-)/2. 

We rewrite Proposition <ref> in the current setting; we emphasize that the shifts appear below because in this case we have the explicit solution (<ref>) to problem (<ref>).


Assume (<ref>). 
Problem (<ref>)-(<ref>) admits a (completely) non-stationary traveling wave if and only if

    α_1,jδ_1,j=v_1,j.

In this case any non-stationary traveling wave ρ has a profile φ of the form

    ϕ(ξ)=(ψ_1(ξ +σ_1), …, ψ_n+1(ξ+σ_n+1)),      ξ∈,

with ℓ_h^± satisfying (i), (ii) and (<ref>) in Proposition <ref> and σ_h∈, h∈𝖧, such that

    c_jσ_1 =c_1σ_j,      j∈𝖩.




By Theorem <ref>, any solution to (<ref>) has the form (<ref>) with σ_h∈, h∈𝖧. Therefore, by Proposition <ref> it only remains to prove that (<ref>) is equivalent to (<ref>)-(<ref>).
Straightforward computations show that in the present case (<ref>) can be written as

    f_j(ℓ_j^+)  ζ_j(t) + f_j(ℓ_j^-)/1+ζ_j(t) = α_1,j f_1(ℓ_1^+)  ζ_1(t) + f_1(ℓ_1^-)/1+ζ_1(t),   t ∈,   j ∈𝖩,

where ζ_h(t) ≐exp z_h(t), for z_h(t) ≐v_h/δ_h  (ℓ_h^+-ℓ_h^-) (c_ht+σ_h), h ∈𝖧.
By Proposition <ref> we have

    either    f_j(ℓ_j^±) = α_1,jf_1(ℓ_1^±),       or    f_j(ℓ_j^±) = α_1,jf_1(ℓ_1^∓).

∙ In the former case, identity (<ref>) is equivalent to

    [ f_j(ℓ_j^+) - f_j(ℓ_j^-) ] [ζ_j(t)-ζ_1(t) ]=0,
       t ∈,   j ∈𝖩.

Since by assumption f_j(ℓ_j^+) ≠ f_j(ℓ_j^-), it must be ζ_j ≡ζ_1, i.e., z_j(t) = z_1(t), namely

    v_j/δ_j  (ℓ_j^+-ℓ_j^-)   c_j = v_1/δ_1  (ℓ_1^+-ℓ_1^-)   c_1,
    
    v_j/δ_j  (ℓ_j^+-ℓ_j^-)  σ_j = v_1/δ_1  (ℓ_1^+-ℓ_1^-)  σ_1,
          ⇔      v_1,j/δ_1,j = f_j(ℓ_j^+) - f_j(ℓ_j^-)/f_1(ℓ_1^+) - f_1(ℓ_1^-) = α_1,j,
    
    σ_j/c_j = σ_1/c_1.


∙ In the latter case, identity (<ref>) is equivalent to

    [ f_j(ℓ_j^+) - f_j(ℓ_j^-) ][ζ_j(t)ζ_1(t)-1]=0,
       t ∈,   j ∈𝖩.

Since by assumption f_j(ℓ_j^+) ≠ f_j(ℓ_j^-), it must be ζ_j  ζ_1 ≡ 1, i.e. z_j(t) = -z_1(t), namely

    v_j/δ_j  (ℓ_j^+-ℓ_j^-)   c_j =- v_1/δ_1  (ℓ_1^+-ℓ_1^-)   c_1,
    
    v_j/δ_j  (ℓ_j^+-ℓ_j^-)  σ_j =- v_1/δ_1  (ℓ_1^+-ℓ_1^-)  σ_1,
          ⇔      v_1,j/δ_1,j = -f_j(ℓ_j^+) - f_j(ℓ_j^-)/f_1(ℓ_1^+) - f_1(ℓ_1^-) = α_1,j,
    
    σ_j/c_j = σ_1/c_1.

In both cases we proved that (<ref>) is equivalent to (<ref>)-(<ref>); this concludes the proof.



Consider conditions (<ref>)_1, (<ref>)_2 and (<ref>). Any two of them implies the third one.



Assume (<ref>). 
Problem (<ref>)-(<ref>) admits a (completely) non-stationary traveling wave satisfying (<ref>) if and only if (<ref>) holds true.
In this case a non-stationary traveling wave satisfies (<ref>) if and only if its end states satisfy (<ref>).



The first part of the statement is just Remark <ref>. In this case, since (<ref>) implies (<ref>), by Proposition <ref> any (completely) non-stationary traveling wave ρ has a profile of the form (<ref>)-(<ref>).

The second part of the statement characterizes the end states. If a non-stationary traveling wave ρ satisfies (<ref>), then (<ref>) holds because of Lemma <ref>. Conversely, if the end states of ρ satisfy (<ref>), then long but straightforward computations show that (<ref>) holds true, and therefore ρ satisfies (<ref>). 






 §.§ The case of linear diffusivities

In this subsection we assume

    D(ρ) ≐ρ.

We notice that D degenerates at 0 and this makes the subject more interesting. In this case problem (<ref>)-(<ref>) reduces to

    δ_hφ_hφ'_h =
    v_h (ℓ^+_h - φ_h) (φ_h - ℓ^-_h),   ξ∈,
    
    φ_h(±∞)=ℓ_h^±.

If ℓ_h^-=0, then the function

    ψ_h(ξ) ≐ℓ_h^+/2(2-e^-v_h/δ_hξ)    if ξ≥-δ_h/v_hln2,
    
    
    0    if ξ<-δ_h/v_hln2,

solves (<ref>) because ℓ_h^- <ℓ_h^+; by (<ref>) we have I_h=(-δ_h/v_hln2,∞). If ℓ_h^->0, then I_h= and the function ψ_h implicitly given by

    [
    2 exp( v_hδ_h ξ) ψ_h(ξ)-ℓ_h^-/ℓ_h^+-ℓ_h^-]^ℓ_h^-
    =
    [
    2 exp( v_hδ_h ξ) ℓ_h^+-ψ_h(ξ)/ℓ_h^+-ℓ_h^-]^ℓ_h^+

solves (<ref>) because ℓ_h^- <ℓ_h^+.
Notice that in both cases ψ_h(0)=(ℓ_h^+ + ℓ_h^-)/2 and all the other solutions are of the form ϕ_h(ξ)=ψ_h(ξ+σ_h) for σ_h ∈.
Hence, any non-stationary traveling wave ρ has a profile φ of the form

    ϕ(ξ)=(ψ_1(ξ +σ_1), …, ψ_n+1(ξ+σ_n+1)),      ξ∈.

In the sequel we prove that the shifts σ_h, h ∈𝖧, satisfy (<ref>), or equivalently

    v_1,j σ_1 =δ_1,j σ_j,      j∈𝖩.



Assume (<ref>). If ℓ_1^+ + ℓ_1^-1, then condition (<ref>) is equivalent to

    v_1,j^2/δ_1,j = 1-ℓ_j^+-ℓ_j^-/1-ℓ_1^+-ℓ_1^-    and       ℓ_j^-  ℓ_j^+ = α_1,j  v_1,j ℓ_1^-  ℓ_1^+.



In the present case, condition (<ref>) becomes
(c_1,jv_1,j - δ_1,j)  ℓ - δ_1,j  k_j = 0 for ℓ∈ (ℓ_j^-,ℓ_j^+):
it is satisfied if and only if both c_1,jv_1,j =  δ_1,j and k_j=0.
The former is equivalent to  (<ref>)_1, the latter is equivalent to (<ref>)_2 by (<ref>)_4, because κ_j=0.


We observe that (<ref>)_1 and (<ref>)_1 imply that c_1,j = δ_1,j/v_1,j > 0; therefore (<ref>) becomes

    ℓ_j^±(1- ℓ_j^±)=α_1,j  v_1,j ℓ_1^±(1- ℓ_1^±),        j ∈𝖩.

As a consequence ρ is either non-degenerate or completely degenerate.

Now, we discuss (completely) non-stationary traveling waves by considering separately the (completely) degenerate and non-degenerate case.
We denote

    Δ_j ≐{α_1,j δ_1,j,   √(δ_1,j),   √(α_1,j δ^2_1,j)},      j ∈𝖩.


Assume (<ref>).
Problem (<ref>)-(<ref>) admits a traveling wave that is both (completely) degenerate and (completely) non-stationary if and only if either (<ref>) holds true or

    [                                             0 <    v_1,j < minΔ_j    or       v_1,j > maxΔ_j,      j ∈𝖩,; v_1,2 (δ_1,2-v_1,2^2)α_1,2 δ_1,2^2-v_1,2^3 =
    … =
    v_1,n+1 (δ_1,n+1-v_1,n+1^2)α_1,n+1 δ_1,n+1^2-v_1,n+1^3. ]


In the first case, problem (<ref>)-(<ref>) has infinitely many of such waves; each of them satisfies (<ref>) and (up to shifts) (<ref>).

In the second case, problem (<ref>)-(<ref>) has a unique (up to shifts) such wave, which does not satisfy (for no shifts) (<ref>). Its end states do not satisfy (<ref>) and are

    ℓ_1^- = 0 = ℓ_j^-,      ℓ_1^+ = v_1,j (δ_1,j-v_1,j^2)/α_1,j δ_1,j^2-v_1,j^3,      ℓ_j^+ = α_1,j δ_1,j (δ_1,j-v_1,j^2)/α_1,j δ_1,j^2-v_1,j^3,      j ∈𝖩.

In both cases, any degenerate non-stationary traveling wave ρ has a profile φ of the form (<ref>) with ψ_h defined by (<ref>) and σ_h∈, h∈𝖧, satisfying (<ref>).



We claim that the existence of a degenerate non-stationary traveling wave is equivalent to the existence of ℓ_h^+∈ (0,1),   h ∈𝖧, such that

    ℓ^+_j=α_1,j δ_1,j/v_1,j ℓ_1^+       and       [ α_1,j δ_1,j^2 - v_1,j^3 ] ℓ_1^+
    + v_1,j[v_1,j^2 - δ_1,j]  = 0,      j∈𝖩.

In fact, by Proposition <ref> the existence of a non-stationary traveling wave is equivalent to condition (𝒯_q), where (<ref>) can be written as (<ref>) by Lemma <ref> and (<ref>) as (<ref>).
Then, (<ref>) and (<ref>) with ℓ_h^-=0, h ∈𝖧, reduce to the relation among the end states

    v_1,j^2/δ_1,j=1-ℓ_j^+/1-ℓ_1^+,      ℓ_j^+   (1 - ℓ_j^+) = α_1,j  v_1,j ℓ_1^+   (1 - ℓ_1^+),      j ∈𝖩.

By (<ref>) we obtain v_1,j^2/δ_1,j = α_1,j  v_1,j ℓ_1^+/ℓ_j^+ and then (<ref>)_1; by plugging (<ref>)_1 into (<ref>)_2 we get (<ref>)_2 and then the claim.

Assume there is a degenerate non-stationary traveling wave; then ℓ_1^+ satisfies (<ref>)_2. As a consequence, we have either α_1,jδ^2_1,j- v^3_1,j=v_1,j^2-δ_1,j=0 or α_1,jδ^2_1,j v^3_1,j for every j ∈𝖩. The former case is equivalent to  (<ref>). In the latter case we can explicitly compute ℓ_1^+ by (<ref>)_2 for any j ∈𝖩 and impose the constraint 0<ℓ_1^+<1, namely,

    0< v_1,j  (δ_1,j-v_1,j^2)α_1,j δ_1,j^2-v_1,j^3  < 1.

A direct computation shows that this is equivalent to (<ref>). In conclusion, either condition (<ref>) or (<ref>) is necessary for the existence of a non-stationary traveling wave with ℓ_1^-=0.

Conversely, assume condition (<ref>). In this case α_1,j δ_1,j^2 =α_1,j  v_1,j^4=v_1,j^3. Then (<ref>)_2 is trivially satisfied for every ℓ_1^+∈ (0,1) and from (<ref>)_1  we deduce ℓ_1^+=ℓ_j^+. Hence, there is an infinite family of non-stationary traveling waves parameterized by ℓ_1^+ ∈ (0,1) and satisfying (<ref>); as a consequence, they do not coincide up to shifts and they all satisfy (up to shifts) the continuity condition (<ref>).

Assume now condition (<ref>). In this case the values for ℓ_1^+ and ℓ_j^+ in (<ref>) are well defined since v^3_1,jα_1,jδ_1,j and they are the unique solution to (<ref>). In particular, condition (ii) in Proposition <ref> is automatically satisfied. By the estimates in (<ref>) we have ℓ_1^+,ℓ_j^+ ∈ (0,1) for j ∈𝖩. Hence, there is a unique (up to shifts) degenerate non-stationary traveling wave and its end states satisfy (<ref>). Furthermore, by Lemma <ref>, condition (<ref>) implies  (<ref>), which is  precluded by (<ref>). Hence, the traveling wave does not satisfy (<ref>).

At last, by Theorem <ref>, any solution to (<ref>) has the form (<ref>).
By (<ref>), that in the present case becomes

    ϕ_j'(c_jξ) = α_1,j c_1,j^2 ϕ_1'(c_1ξ)       for a.e. ξ∈, j ∈𝖩,

and the regularity of ψ_h defined in (<ref>), we have

    1/c_j[ δ_j/v_j ln 2 + σ_j ]
    =
    1/c_1[ δ_1/v_1 ln 2 + σ_1 ],

which is equivalent to (<ref>) because c_1,j = δ_1,j/v_1,j.


The following result treats the non-degenerate case.


Assume (<ref>).
Problem (<ref>)-(<ref>) admits a non-degenerate (completely) non-stationary traveling wave if and only if condition (<ref>) is satisfied. In this case any non-degenerate non-stationary traveling wave satisfies (up to shifts) (<ref>); moreover, it has a profile φ of the form (<ref>) with ψ_h implicitly defined by (<ref>) and σ_h∈, h∈𝖧, satisfying (<ref>).


Assume that there is a non-degenerate non-stationary traveling wave; 
then ℓ_h^-  0 and 1 ℓ_h^+ + ℓ_h^-, h ∈𝖧. Moreover, by Proposition <ref>, condition (𝒯_q) is satisfied, where (<ref>) becomes (<ref>) by Lemma <ref> and (<ref>) is (<ref>).
When dividing (<ref>) by (<ref>)_2 we obtain

    (1-ℓ_j^+)  ℓ_1^- = (1-ℓ_1^+)  ℓ_j^-       and       (1-ℓ_j^-)  ℓ_1^+ = (1-ℓ_1^-)  ℓ_j^+,      j ∈𝖩.

By adding the above relations we have ℓ_1^- + ℓ_1^+ = ℓ_j^- + ℓ_j^+, hence

    0 = ℓ_1^+ - ℓ_j^+ + ℓ_1^- - ℓ_j^-
    = ℓ_1^+ - 1 + (1-ℓ_1^+)  ℓ_j^-/ℓ_1^- + ℓ_1^- - ℓ_j^-
    
    
    = 1-ℓ_1^+-ℓ_1^-/ℓ_1^-  (ℓ_j^- - ℓ_1^-).

It is now easy to conclude that (<ref>) is satisfied and then also (<ref>) holds true by (<ref>). At last, the traveling wave satisfies (up to shifts) (<ref>) by Remark <ref>.

Conversely, assume (<ref>). 
Then (<ref>) and (<ref>) write

    ℓ_j^+ + ℓ_j^- = ℓ_1^+ + ℓ_1^-,       ℓ_j^-  ℓ_j^+ =ℓ_1^-  ℓ_1^+,       ℓ_j^±(1- ℓ_j^±)=ℓ_1^±(1- ℓ_1^±),       j ∈𝖩.

The same computations as before give that if we impose ℓ_h^-  0 and 1 ℓ_h^+ + ℓ_h^-, h ∈𝖧, then the above conditions are equivalent to (<ref>); the existence of infinitely many non-degenerate non-stationary traveling waves satisfying (<ref>) easily follows.

At last, by Theorem <ref>, any solution to (<ref>) has the form (<ref>).
Fix j ∈𝖩. 
By (<ref>) we have (<ref>), namely

    ψ_j(c_jt+σ_j) = ψ_1(c_1t+σ_1),      t ∈.

This identity together with (<ref>) and (<ref>) imply

    [
    2 exp( v_jδ_j  (c_jt+σ_j) ) ψ_1(c_1t+σ_1)-ℓ^-/ℓ^+-ℓ^-]^ℓ^-
    =
    [
    2 exp( v_jδ_j  (c_jt+σ_j) ) ℓ^+-ψ_1(c_1t+σ_1)/ℓ^+-ℓ^-]^ℓ^+,
    
       [
    2 exp( v_1δ_1  (c_1t+σ_1) ) ψ_1(c_1t+σ_1)-ℓ^-/ℓ^+-ℓ^-]^ℓ^-
    =
    [
    2 exp( v_1δ_1  (c_1t+σ_1) ) ℓ^+-ψ_1(c_1t+σ_1)/ℓ^+-ℓ^-]^ℓ^+.

By dividing the above equalities and taking the logarithm we get

    [ v_jδ_j  (c_jt+σ_j) - v_1δ_1  (c_1t+σ_1) ] ℓ^- = [ v_jδ_j  (c_jt+σ_j) - v_1δ_1  (c_1t+σ_1) ] ℓ^+,      t ∈.

Since ℓ^- ℓ^+ and c_1,j = δ_1,j/v_1,j, the above equality is equivalently to (<ref>).






§ APPLICATION TO THE CASE OF A LOGARITHMIC FLUX, 


In this section we assume (<ref>) for some constants v_h,δ_h>0,  D ≐ 1 and the logarithmic flux <cit.> defined by

    f(ρ) ≐ -ρln(ρ)

for ρ∈(0,1] with f(0)=0 by continuity; in the following we simply write ρln (ρ) for ρ∈ [0,1]. We use the notation introduced in (<ref>); then, in the present case the diffusivity D_h coincides with the anticipation length δ_h of <cit.>, see Section <ref>. As in Section <ref>, we focus on the case m=1 and do not mention in the following these assumptions on f_h, D_h and m.  

Condition (<ref>) becomes

    0≤ℓ_h^- < ℓ_h^+≤1       and       c_h=-v_hℓ^+_hln(ℓ^+_h)-ℓ^-_hln(ℓ^-_h)/ℓ^+_h-ℓ^-_h.

Moreover we have, for h ∈𝖧,

    g_h(ℓ)=
    v_hℓ[ℓ^+_hln(ℓ^+_h)-ℓ^-_hln(ℓ^-_h)/ℓ^+_h-ℓ^-_h - ln(ℓ)],
    
       g_h(ℓ) - g_h(ℓ_h^±)=
    v_h [ (ℓ-ℓ_h^-)ℓ^+_hln(ℓ^+_h)+(ℓ_h^+-ℓ)ℓ^-_hln(ℓ^-_h)/ℓ^+_h-ℓ^-_h - ℓln(ℓ)].

Therefore (<ref>) becomes

    ϕ_h'(ξ)=v_h/δ_h[ [ϕ_h(ξ) - ℓ_h^-] ℓ^+_h ln(ℓ^+_h) + [ℓ_h^+ - ϕ_h(ξ)] ℓ^-_h ln(ℓ^-_h)/ℓ^+_h - ℓ^-_h
    - ϕ_h(ξ) ln(ϕ_h(ξ)) ],

for ξ∈. Let :[0,e^-1] → [0,e^-1] and :[0,e^-1] → [e^-1,1] be the inverse functions of the restrictions f_ℓ and f_r of f to [0, e^-1] and [e^-1, 1], respectively.

We first consider the case of stationary waves. We define the intervals

    ℒ_j^0≐
    [0, e^-1)   if α_1,j  v_1,j≤ 1,
    
    [0, (e^-1α_1,j^-1  v_1,j^-1) )   if α_1,j  v_1,j > 1,
          j ∈𝖩.



Problem (<ref>)-(<ref>) admits infinitely many stationary traveling waves; their end states are characterized by the conditions

    ℓ_1^- ∈⋂_j ∈𝖩ℒ_j^0,      ℓ_1^+ = (-ℓ_1^-ln (ℓ_1^-)),
       ℓ_j^- = (-α_1,j  v_1,jℓ_1^-ln (ℓ_1^-) ),      ℓ_j^+ = (-α_1,j  v_1,jℓ_1^-ln (ℓ_1^-)),      j ∈𝖩.

Moreover, up to shifts, any stationary traveling wave satisfies (<ref>).



The first part of the proposition follows from Theorem <ref>.
Indeed, conditions (<ref>)_1 and  (<ref>) are satisfied if and only if for any h∈𝖧 and j∈𝖩 

    ℓ_h^- ∈ [0, e^-1),      ℓ_h^-ln(ℓ_h^-)=ℓ_h^+ln(ℓ_h^+),      ℓ_j^-  ln( ℓ_j^-)= α_1,j  v_1,j ℓ_1^-  ln( ℓ_1^-).

Hence ℓ_1^+=(-ℓ_1^-ln(ℓ_1^-)) and it is sufficient to determine ℓ_j^±.  Observe that the definition of ℒ_j^0 guarantees that they can be uniquely computed. 
At last, the latter part of the proposition follows by the proof of Proposition <ref> since e^-1∈S≐⋂_h∈𝖧(ℓ_h^-,ℓ_h^+) ∅.


In the following  we discuss the existence of non-stationary traveling waves. Since m=1, by Lemma <ref> this is equivalent to assume that  the traveling wave is completely non-stationary. By (<ref>)_2 we deduce

    c_1,j = v_1,j ℓ_1^+ln(ℓ_1^+)-ℓ_1^-ln(ℓ_1^-)/ℓ_j^+ln(ℓ_j^+)-ℓ_j^-ln(ℓ_j^-) ℓ_j^+-ℓ_j^-/ℓ_1^+-ℓ_1^-.

The following result translates Theorem <ref> to the current framework. We define the intervals

    ℒ_j^c≐
    [0,1]   if α_1,j  v_1,j≤ 1,
    
    
    [0,1] ∖((e^-1α_1,j^-1  v_1,j^-1) ,  (e^-1α_1,j^-1  v_1,j^-1))    if α_1,j  v_1,j > 1,
          j ∈𝖩.



Problem (<ref>)-(<ref>) admits a (completely) non-stationary traveling wave if and only if the following condition holds.


  (𝒯_l)  There exist ℓ_1^±∈ [0,1] with ℓ_1^-<ℓ_1^+ such that:



  (i)  ℓ_1^-ln(ℓ_1^-)ℓ_1^+ ln(ℓ_1^+);


  (ii) ℓ_1^±∈⋂_j ∈𝖩ℒ_j^c;


  (iii) for any j ∈𝖩 we have

    δ_1,j[ g_j(ℓ)- g_j(ℓ_j^-) ] = A_1,j  c_1,j[ g_1( ℓ+k_j/A_1,j)-g_1(ℓ_1^-) ],      ℓ∈ (ℓ^-_j, ℓ^+_j),

where g_h is given in (<ref>), c_1,j in (<ref>), A_1,j in (<ref>)_2 and k_j in (<ref>)_3, with ℓ_j^± being solutions to

    ℓ_j^±ln(ℓ_j^±)=α_1,jv_1,jL_1,j^±ln(L_1,j^±).





 The proof consists in showing that, in the present case, (𝒯) of Theorem <ref> is equivalent to (𝒯_l). The first two items in  (𝒯) and  (𝒯_l) are clearly equivalent.
 It remains to discuss the third one. Condition (<ref>) is equivalent to 

    δ_1,j[g_j(ℓ_j(c_jξ))- g_j(ℓ_j^-)] = A_1,j  c_1,j[ g_1(ϕ_1(c_1ξ))-g_1(ℓ_1^-)],      ξ∈,

where ϕ_1 is a solution to (<ref>)-(<ref>) and ℓ_j(ξ) ≐ A_1,jϕ_1(c_1,jξ)-k_j for c_1,j in (<ref>), A_1,j in (<ref>)_2 and k_j in (<ref>)_3. By Theorem <ref>, ϕ_1 is strictly increasing and so is the function ℓ_j. Put ℓ≐ℓ_j(c_jξ). Hence ℓ∈ (ℓ_j^-,  ℓ_j^+), by Lemma <ref>, and then (<ref>) is equivalent to (<ref>).


In the following we focus on the case of (completely) non-stationary traveling waves with ℓ_h^-=0 for some h ∈𝖧.


Assume that problem (<ref>)-(<ref>) admits a traveling wave. The following statements are equivalent:


  (i)
ℓ_1^- = 0;

  (ii)
ℓ_j^- = 0 for all j ∈𝖩;

  (iii)
there exists j∈𝖩 such that ℓ_ j^- = 0.




First, we prove that (i) implies (ii). 
Fix j ∈𝖩.
Since ℓ_1^-=0, then condition (<ref>) implies that either ℓ_j^-=0 or ℓ_j^+=1, for j ∈𝖩. Assume by contradiction that ℓ_j^+=1. 
Since c_1,j<0, condition (<ref>) becomes 

    ℓ_j^-ln(ℓ_j^-)=α_1,j  v_1,jℓ_1^+ln(ℓ_1^+).

Therefore, by (<ref>), (<ref>)_2 and  (<ref>)_3 we have that

    c_1,j = - v_1,j ℓ_1^+ln(ℓ_1^+)/ℓ_j^-ln(ℓ_j^-) 1-ℓ_j^-/ℓ_1^+
    = - 1-ℓ_j^-/α_1,j ℓ_1^+,      A_i,j  = - 1-ℓ_j^-/ℓ_1^+,
       k_j =  -1.

Condition (<ref>) can be written as

    ℓln(ℓ) - v_1,j (1-ℓ) [ α_1,j ℓ_1^+ln(ℓ_1^+)/1-ℓ_j^-
    + 1-ℓ_j^-/α_1,j δ_1,j ℓ_1^+ ln( 1-ℓ/1-ℓ_j^-) ]=0,

for ℓ∈ (ℓ_j^-, 1).
By differentiating the above equation three times we obtain

    -v_1,j(1-ℓ_j^-)/α_1,jδ_1,jℓ_1^+(1-ℓ)^2 = 1/ℓ^2,      ℓ∈ (ℓ_j^-, 1).

This is a contradiction because the two sides have opposite sign. This proves (ii).

Since the implication (ii) ⇒ (iii) is obvious, it remains to show that (iii) ⇒ (i). Let ℓ_ j^-=0 for some j∈𝖩. By (<ref>) it follows that either ℓ_1^-=0 or ℓ_1^+=1. In the latter case by arguing as above it is easy to obtain a contradiction and then (iii) follows.


At last, we give a result which is similar to the one given in Proposition <ref>. We denote 

    Δ_j ≐{α_1,j δ_1,j, √(δ_1,j)},      j∈𝖩.

By Lemma <ref> we have either ℓ_h^-=0, h ∈𝖧, or  ℓ_h^-0, h ∈𝖧.
Below we consider the first case.


Problem (<ref>)-(<ref>) admits a (completely) non-stationary traveling wave with ℓ_h^-=0, h ∈𝖧, if and only if either (<ref>) holds true or

    [                                         0 <    v_1,j < minΔ_j    or       v_1,j > maxΔ_j,      j ∈𝖩,; [α_1,2δ_1,2/v_1,2]^δ_1,2/v_1,2^2 - δ_1,2 =
    … =
    [α_1,n+1δ_1,n+1/v_1,n+1]^δ_1,n+1/v_1,n+1^2 - δ_1,n+1. ]


In the first case, problem (<ref>)-(<ref>) has infinitely many of such waves; each of them satisfies (<ref>) and (up to shifts) (<ref>).

In the second case, problem (<ref>)-(<ref>) has a unique (up to shifts) such wave and such wave, which does not satisfy (for no shifts) (<ref>). Its end states are

    ℓ_1^-=0=ℓ_j^-,      ℓ_1^+
    =
    [α_1,jδ_1,j/v_1,j]^δ_1,j/v_1,j^2 - δ_1,j,
       ℓ_j^+ = [α_1,jδ_1,j/v_1,j]^v_1,j^2/v_1,j^2 - δ_1,j,        j ∈𝖩,

and do not satisfy (<ref>).


 
Fix j ∈𝖩.
Since c_1,j > 0, by (<ref>) the formulas in (<ref>) and (<ref>) become

    c_1,j=v_1,j ln(ℓ_1^+)/ln(ℓ_j^+),    
    A_1,j=α_1,j  v_1,j ln(ℓ_1^+)/ln(ℓ_j^+),    
    k_j=0=κ_j,
    
    g_h(ℓ) = v_h  ℓ ln(ℓ_h^+/ℓ),    
    g_h(0)=0.

Hence (<ref>) can be written as 

    ℓ_j^+ ln(ℓ_j^+) = α_1,j  v_1,j ℓ_1^+ ln(ℓ_1^+)

and therefore (<ref>) becomes

    [ δ_1,j - v_1,j ℓ_j^+/α_1,j ℓ_1^+] ln( ℓ_j^+/ℓ) = 0,      ℓ∈ (0, ℓ_j^+),

namely

    ℓ_j^+ = α_1,j δ_1,j/v_1,j ℓ_1^+.

System (<ref>)-(<ref>) admits a solution if and only if either (<ref>) or (<ref>) holds true. In  the former case, (<ref>)-(<ref>) has infinitely many solutions and they satisfy (<ref>); in the latter, the unique solution of (<ref>)-(<ref>) is (<ref>)_2,3. We examine separately these cases.

Assume (<ref>). In this case condition (𝒯_l) of Proposition <ref> with ℓ_1^-=0=ℓ_j^- is equivalent to ℓ_1^+=ℓ_j^+ ∈ (0,1),   j ∈𝖩, and then there are infinitely many traveling waves. They all satisfy (<ref>) by Remark <ref>.

Assume (<ref>). In this case condition (𝒯_l) of Proposition <ref> with ℓ_1^-=0=ℓ_j^- is equivalent to ℓ_h^+ ∈ (0,1), h ∈𝖧, satisfying (<ref>)-(<ref>), namely to (<ref>)-(<ref>).
In particular, (<ref>)_1, (<ref>) imply that ℓ_j^+ and ℓ_1^+ are distinct, namely they do not satisfy (<ref>).
Moreover, by Remark <ref> the traveling wave does not satisfies (<ref>).

At last, the reverse implications are direct consequences of previous discussion about the solutions of (<ref>)-(<ref>) and then the proof is complete.





§ PROOF OF THEOREM <REF>


Let ℓ^±_h ∈ [0, 1] with ℓ^-_hℓ^+_h.  We introduce the change of variable

    r_h ≐ℓ^+_h-ρ_h/ℓ^+_h-ℓ^-_h,

which implies ρ_h = ℓ^+_h - (ℓ^+_h-ℓ^-_h)   r_h,
ρ_h,t = -(ℓ^+_h - ℓ^-_h)   r_h,t and
ρ_h,x = -(ℓ^+_h - ℓ^-_h)   r_h,x.
Consequently, equation (<ref>) can be written

    r_h,t + G_h(r_h)_x = (E_h(r_h)   r_h,x)_x,

where

    G_h(r_h) ≐ -f_h(ℓ^+_h - (ℓ^+_h-ℓ^-_h)   r_h)-f_h(ℓ^+_h)/ℓ^+_h -ℓ^-_h,      E_h(r_h)≐ D_h(ℓ^+_h - (ℓ^+_h-ℓ^-_h)   r_h).

Furthermore,  equation (<ref>) has a wavefront solution ψ_h from 1 to 0 with wave speed θ_h  if and only if equation (<ref>) has a wavefront solution ϕ_h from ℓ^-_h to ℓ^+_h with the same speed. Notice that ψ_h satisfies the equation

    (E_h(ψ_h)ψ'_h)' +(θ_h - G'_h(ψ_h)) ψ'_h=0

and  ϕ_h is obtained by ψ_h by the change of variable (<ref>), i.e.

    ϕ_h(ξ) = (ℓ^-_h -ℓ^+_h)  ψ_h(ξ) + ℓ^+_h,      ξ∈ℝ.


We discuss now  the existence of a wavefront solution r_h(t,x)=ψ_h(x-θ_h t+σ_h)=ψ_h(ξ) of (<ref>).
In order to make use of <cit.>, we only need to show that

    -G_h(r_h)>-r_h  G_h(1),      r_h ∈ (0,1).

By the definition of G_h we have

    -r_h   G_h(1) = -r_h  f_h(ℓ^+_h)-f_h(ℓ^-_h)/ℓ^+_h-ℓ^-_h.

Then, inequality (<ref>) is equivalent to

    f_h(ℓ^+_h) - (f_h(ℓ^+_h)-f(ℓ^-_h)) r_h < f_h(ℓ^+_h - (ℓ^+_h-ℓ^-_h)   r_h),       for r_h ∈ (0,1),

if and only if ℓ_h^-<ℓ_h^+. By the strict concavity of f_h the last inequality is satisfied and then, by
 <cit.>, we deduce the existence of wavefront solutions ψ_h  from 1 to 0 for (<ref>). The wave speed, in this case, is  θ_h ≐ G_h(1).
Furthermore, the profile ψ_h is unique up to shifts and, if ψ_h(0)≐ν for some 0<ν< 1, then

    ψ_h(ξ)=1    for ξ≤ν_h^-,
    
    ∫_ψ_h(ξ)^νE_h(s)/-G_h(s)+s   G_h(1)=ξ   for ν_h^-<ξ<ν_h^+ ,
    ψ_h(ξ)=0    for ξ≥ν_h^+,

where

    ν_h^+ ≐∫_0^νE_h(s)/-G_h(s)+ s   G_h(1)  d s,      ν_h^- ≐ - ∫_ν^1E_h(s)/-G_h(s)+ s   G_h(1)  ds.

Notice that, by differentiating (<ref>) in the interval (ν_h^-, ν_h^+), we obtain that

    E_h(ψ_h(ξ))/G_h(ψ_h(ξ))-ψ_h(ξ)   G_h(1) ψ_h'(ξ)=1,      ξ∈ (ν_h^-, ν_h^+),

which implies ψ_h'<0 in (ν_h^-, ν_h^+) because of (<ref>).

Consider now ϕ_h defined in (<ref>); it satisfies (<ref>) with I_h=(ν_h^-, ν_h^+) and ϕ_h'>0 in I_h. Also condition (<ref>) is true and ϕ_h ∈2(I_h, (ℓ^-_h, ℓ^+_h) by the regularity of D_h and f_h.

Now it remains to consider the boundary conditions of ϕ_h' at the extrema of I_h in the different cases. We have the following.



  (i) Assume ℓ_h^-=0=D_h(0). We show that

    ν_h^-= - ∫_ν^1E_h(s)/-G_h(s)+ s   G_h(1)  ds >-∞.

To prove (<ref>), notice that E_h(1)=D_h(0)=0 and that -G_h(s)+ sG_h(1) → 0 as s→ 1^-. In addition, by means of the strict concavity of f_h we obtain that

    lim_s→1^-E'_h(s)/-G_h'(s)+ G_h(1)=E'_h(1)/-G'_h(1)+ G_h(1)=-ℓ_h^+   D_h'(0)/-f_h'(0)+
    f_h(ℓ_h^+)/ℓ_h^+≥ 0

 and then, by applying de l'Hospital Theorem we prove condition (<ref>). Moreover, by condition (<ref>), we get
 
    lim_ξ↓ν_h^-ψ_h'(ξ)=-f_h'(0)-f_h(ℓ_h^+)/ℓ_h^+/ℓ_h^+   D_h'(0)   if  D_h'(0)>0,
    
    -∞   if  D_h'(0)=0.

By applying (<ref>) we conclude that ϕ_h(ξ)=ℓ_h^- for ξ≤ν_h^- and the estimates in (<ref>) are satisfied.
Furthermore, by the change of variables (<ref>), we obtain that

    lim_ξ↓ν_h^-D_h(ϕ_h(ξ))ϕ_h^'(ξ)=lim_ξ↓ν_h^--ℓ_h^+E_h(ψ_h(ξ))ψ_h^'(ξ)

and hence, by (<ref>), we deduce (<ref>).


  (ii) Assume 1-ℓ_h^+=0=D_h(1). With a similar reasoning as in (i) we prove that ν_h^+>-∞. In fact, E_h(0)=D_h(1)=0 and -s   G_h(s)+s   G_h(1) → 0 as s → 0^+. Moreover
 
    lim_s→ 0^+E'_h(s)/-G_h'(s)+ G_h(1)=E'_h(0)/-G'_h(0)+ G_h(1)=( 1-ℓ_h^- )  D_h'(1)/f_h'(1)+
    f_h(ℓ_h^-)/1-ℓ_h^-≥ 0

 and again, by applying de l'Hospital Theorem we prove that ν_h^+>-∞. Moreover, by the estimate (<ref>), we have that

    lim_ξ↑ν^hψ_h'(ξ)=f_h(ℓ_h^-)/1-ℓ_h^-+f_h'(1)/(ℓ_h^- -1)  D_h'(1)   if  D_h'(1)<0,
    
    -∞   if  D_h'(1)=0.

By applying (<ref>) we conclude that ϕ_h(ξ)=ℓ_h^+ for ξ≥ν_h^+ and the estimates in (<ref>) are satisfied; by (<ref>) and (<ref>) we derive (<ref>).


  (iii)  In all the other cases it is easy to show that I_h=ℝ and again the slope condition (<ref>) can be obtained by the estimate (<ref>).





§ ACKNOWLEDGMENTS

The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). They were supported by the Project Macroscopic models of traffic flows: qualitative analysis and implementation, sponsored by the University of Modena and Reggio Emilia.
The first author was also supported by the project Balance Laws in the Modeling of Physical, Biological and Industrial Processes of GNAMPA.



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