Diffusive stability against nonlocalized perturbations of planar wave trains in reaction-diffusion systems
Björn de Rijk Björn Sandstede






Diffusive stability against nonlocalized perturbations of planar wave trains in reaction-diffusion systems
    Björn de Rijk[, Institut für Analysis, Dynamik und Modellierung, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany] Björn Sandstede[, Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA. Sandstede was partially supported by the NSF through grant DMS-1714429.]
    December 30, 2023
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Planar wave trains are traveling wave solutions whose wave profiles are periodic in one spatial direction and constant in the transverse direction. In this paper, we investigate the stability of planar wave trains in reaction-diffusion systems. We establish nonlinear diffusive stability against perturbations that are bounded along a line in ℝ^2 and decay exponentially in the distance from this line. Our analysis is the first to treat spatially nonlocalized perturbations that do not originate from a phase modulation. We also consider perturbations that are fully localized and establish nonlinear stability with better decay rates, suggesting a trade-off between spatial localization of perturbations and temporal decay rate. Our stability analysis utilizes pointwise estimates to exploit the spatial structure of the perturbations. The nonlocalization of perturbations prevents the use of damping estimates in the nonlinear iteration scheme; instead, we track the perturbed solution in two different coordinate systems.

Keywords. Nonlinear stability, pointwise estimates, nonlocalized perturbations, planar reaction-diffusion systems, periodic travelling waves




§ INTRODUCTION


In this paper, we investigate the stability of spatially periodic planar travelling waves. Consider a planar reaction-diffusion system of the form

    u_t = D(u_xx+u_yy) + f(u),    (x,y)∈ℝ^2,    t ≥ 0,    u∈ℝ^n,

where n ∈, D ∈^n × n is a symmetric, positive-definite matrix, and f ^n →^n is a C^3-smooth nonlinearity. We are interested in planar travelling-wave solutions to (<ref>) of the form u(x,y,t)=u_∞(kx-ω t), where the profile u_∞(ζ) is C^3-smooth and periodic in ζ with period 1, k ∈ denotes the spatial wave number, and ω∈ is the temporal frequency of the travelling wave. From now on, we use the term wave train to refer to spatially-periodic travelling waves. We note that the terms “rolls” and “stripes” are also used in literature to refer to planar wave trains.

Our goal is to determine whether, and in what sense, the planar wave train u(x,y,t)=u_∞(kx-ω t) is stable under perturbations of the initial condition u(x,y,0)=u_∞(kx). Part of our motivation stems from the case of planar spiral waves that resemble planar wave trains in the far field: understanding the stability of wave-train solutions to (<ref>) is a first step towards any nonlinear stability analysis of planar spiral waves.

Before discussing the nonlinear stability of wave trains for the planar system (<ref>), we review the relevant results for the spatially one-dimensional case. Note that the function u(x,t)=u_∞(kx-ω t) is also a wave-train solution to the one-dimensional version

    u_t = D u_xx + f(u),    x∈ℝ,    t ≥ 0,    u ∈^n,

of (<ref>). Throughout, we will assume that the wave train is spectrally stable and refer to <ref> for details on what this assumption entails. We then consider initial conditions of the form

    ũ(x,0) = u_∞(kx+ϕ_0(x)) + v_0(x),     ϕ_0(x)→ϕ_±x→±∞,

where the perturbation v_0 is sufficiently small in an appropriate function space, so that we change the phase, but not the wave number, of the wave train at time t=0. Let ũ(x,t) denote the associated solution to (<ref>): we may then ask whether ũ(x,t) converges in an appropriate sense to u_∞(kx-ω t), or a translate, as time t goes to infinity.

More generally, we can attempt to write the solution in the form

    ũ(x,t) = u_∞(kx+ϕ(x,t)-ω t) + .

For the case |ϕ_+-ϕ_-| ≪ 1, it was shown in <cit.> that (<ref>) holds for a function ϕ(x,t) that has an asymptotically self-similar profile as t →∞: indeed, ϕ(x,t) converges to a moving Gaussian if ϕ_+=ϕ_- and to a moving error function with amplitude ϕ_+-ϕ_- in the case where 0<|ϕ_+-ϕ_-|≪1. Similar results, though without the explicit asymptotics, were also shown in <cit.> using different methods – see Remark <ref> below for more details. The results in <cit.> were complemented with explicit asymptotics in <cit.>, recovering the results of <cit.> for the case 0 < |ϕ_+ - ϕ_-| ≪ 1. The restriction that |ϕ_+-ϕ_-| is small was recently removed in <cit.>. We emphasize that, although the initial phase off-set ϕ_0 can be nonlocalized, the perturbation v_0 in (<ref>) has to be localized in all the aforementioned papers, that is, we need to assume that v_0(x) → 0 sufficiently rapidly as x →±∞.

In this paper, we examine the nonlinear stability of planar wave trains u(x,y,t)=u_∞(kx-ω t) that satisfy (<ref>). We note that increasing the spatial dimension from one to two improves the decay properties on the linear level: for instance, the solutions to the heat equation

    u_t = Δ_x u,   x∈^d,    t ≥ 0,    u ∈^n,

decay pointwise with rate t^-d/2 so that increasing the spatial dimension d∈ yields faster decay in time. This leads to the natural question whether the additional decay can be exploited to allow for a larger – or different – class of initial conditions. In this paper, we answer this question in the affirmative by proving that planar wave trains are stable against a class of nonlocalized perturbations v_0(x,y) rendering our initial conditions complementary to those considered in the literature for one spatial dimension – see Remark <ref> below for more details.

More specifically, given initial conditions of the form

    ũ(x,y,0) = u_∞(kx) + v_0(x,y),

we prove that the resulting solution ũ(x,y,t) to (<ref>) decays pointwise with rate t^-1/2 like a diffusive Gaussian if v_0(x,y) is bounded along an arbitrary, but fixed, line in ℝ^2 and decays exponentially in the distance from this line. We note that the obtained decay is optimal for this class of perturbations – see Remark <ref> below – although it is slower than one would intuitively expect suggesting a trade-off between spatial localization of the initial perturbation and temporal decay rate. Indeed, our second result recovers the expected pointwise decay with rate t^-1 provided v_0(x,y) is sufficiently localized in all spatial directions in ^2. Although this second theorem provides an expected result, it seems not to be present in the literature at the moment – see Remark <ref> below.

The proofs of our main results depend on pointwise estimates that allow us to exploit the specific spatial structure of the perturbations. Therefore, we extend the pointwise Green's function estimates  for wave trains proved in <cit.> in one spatial dimension to the planar case. However, the nonlinear stability analysis in <cit.> – which is, in turn, based on <cit.> – does not extend to our planar case: the L^2-localization of the initial perturbation is utilized in <cit.> in a crucial way in their nonlinear damping estimates, which are necessary to close the nonlinear iteration arguments presented in <cit.>. The main challenge in our proof is to replace these damping estimates: we rely instead on tracking the perturbed solution in two different coordinate systems, where one of these coordinates is used to correct for the phase shift of the solution relative to the original wave train.

We now comment briefly on open problems. A major open problem in the one-dimensional case is the long-time dynamics of initial conditions of the form ũ(x,0) = u_∞(k_0(x)x), which correspond to perturbations of the wave number instead of just the phase. In the planar case, not much is known about the linear and nonlinear stability of spiral waves. Moreover, it is unexplored whether, and in what form, planar wave trains are stable against an initial change in phase, that is whether, and in what form, stability holds for initial conditions of the form ũ(x,y,0) = u_∞(kx + ϕ_0(x,y)) + v_0(x,y) with v_0 possibly nonlocalized. Finally, one expects even higher decay rates in the stability analysis of wave trains in spatial dimensions d ≥ 3, and one could therefore attempt studying even larger classes of initial conditions in this setting.

 
There are various methods to prove nonlinear stability results of wave-train solutions to reaction-diffusion systems in one spatial dimension. The proofs in <cit.> are based on a decomposition of phase and amplitude variables as in <cit.>, mode filters to separate critical (translational) modes from noncritical exponentially decaying modes, and on the renormalization group techniques developed in <cit.>. On the other hand, in <cit.>, L^p-estimates on the Green's function (again obtained by separating critical from noncritical modes) are applied in a nonlinear iteration scheme that was originally devised in shock-wave theory. The set-up in <cit.> follows <cit.>, but employs pointwise Green's function estimates instead. The use of pointwise Green's function estimates was introduced in <cit.> and further extended in <cit.> to the setting of wave trains: it has the advantage of resulting in detailed information on both temporal and spatial decay. Finally, in <cit.>, a normal form about the wave train was constructed that arises as a conjugation of the reaction-diffusion system (<ref>) with a lattice dynamical system of discrete phase equations. The normal form exhibits a conservation law associated with the translational symmetry, which can be exploited to gain additional decay in the nonlinear stability argument.



Although this paper seems to be the first to consider the nonlinear stability of wave trains in planar reaction-diffusion systems (<ref>), the methods for proving nonlinear stability in one spatial dimension – see Remark <ref> – have been applied in higher space dimensions to different systems. For instance, the renormalization approach from <cit.> is used in <cit.> to prove that the planar Swift-Hohenberg equation admits nonlinearly stable wave-train solutions. In addition, the methods in <cit.> are applied in higher space dimensions in <cit.> to prove nonlinear stability results for wave trains in viscous systems of conservation laws. We emphasize that, unlike in the current paper, localized perturbations were considered in these references.


 
In this paper, we consider initial conditions of the form (<ref>), where the perturbation v_0(x,y) is bounded along a line in ℝ^2 and decays exponentially in the distance from this line. All initial conditions for one spatial dimension considered in the literature – see <cit.> – are of the form (<ref>), where the perturbation v_0(x) is sufficiently localized and the initial phase off-set ϕ_0(x) satisfies ϕ_0(x) →ϕ_± as x →±∞. This implies that, in the limit x →±∞, the initial condition ũ(x,0) converges to the translate u_∞(kx + ϕ_±) of the wave train. In our case, the initial condition (<ref>) does not (necessarily) converge to a translate of the wave train when letting |(x,y)| →∞ over some line in ^2. Indeed, we allow for instance for the perturbation v_0(x,y) = ϵ e^-y^2 with 0 < ϵ≪ 1. Thus, the nonlocalization of the perturbations in this paper does not originate from a phase modulation, and our class of initial conditions is therefore complementary to those present in literature.




§ MAIN RESULTS
 

In this paper, we establish nonlinear diffusive stability of the planar wave-train solution u(x,y,t) = u_∞(kx-ω t) to (<ref>) against a class of spatially localized perturbations and a class of spatially nonlocalized perturbations under the assumption that the wave train is spectrally stable. Our nonlinear stability results and the associated spectral assumptions are most naturally formulated by switching to a co-moving frame = kx - ω t, which yields a stationary wave-train solution u(,y,t) = u_∞() to

    u_t = D(k^2 ∂_ + ∂_yy)u + ω∂_ u + f(u),    (,y) ∈^2,    t ≥ 0,    u ∈^n,

Before stating our nonlinear stability results, we disclose what spectral stability entails.



 §.§ Spectral stability


The linearization of (<ref>) about u(,y,t) = u_∞() is

    u_t =  u,      u = D(k^2 ∂_ + ∂_yy) u + ω∂_ u + f'(u_∞()) u,

where  is a sectorial operator on L^2(^2,^n) – see <cit.> and note that  is elliptic as D is a symmetric positive-definite matrix. We apply the Fourier transform , given by

    (,ν_y) = 1/√(2π)∫_ e^-iν_y y u(,y)dy,

in the transverse coordinate y to the linearization (<ref>) and obtain

    _t = _ν_y,     _ν_y  = D(k^2 ∂_ - ν_y^2)  + ω∂_ + f'(u_∞()) ,

where _ν_y is a sectorial operator on L^2(,^n) for each ν_y ∈. The solution operator is represented via the inverse Fourier transform as

    [e^ t u](,y) = 1/√(2π)∫_ e^iν_y y e^_ν_y t(,ν_y)dν_y.

Since _ν_y is a periodic differential operator on L^2(,^n), we can apply the Bloch transform to (<ref>) – see <cit.>. The composition  of the Bloch and Fourier transforms is given by

    (,ν) = ∑_j ∈ e^2π i j _2(u)(ν_x + 2π j,ν_y),     _2(u)(ν) = 1/2π∫_^2 e^-iν·ς u(,y)d dy,

where _2 is the Fourier transform on ^2, · denotes the dot product, ν = (ν_x,ν_y) and ς = (,y). Applying the Fourier-Bloch transform to (<ref>) yields

    _t = L_ν,      L_ν = D(k^2 (∂_ + iν_x)^2 - ν_y^2)  + ω (∂_ + iν_x)  + f'(u_∞()) ,

where L_ν is a sectorial operator on L_^2([0,1],^n) with compact resolvent for each ν∈Ω := [-π,π] ×. The solution operator is represented via the inverse Fourier-Bloch transform as

    [e^ t u](,y) = 1/2π∫_Ω e^iν·ς e^L_ν t(ν,)dν.

Furthermore, we obtain the spectral decomposition

    σ() = ⋃_ν∈Ωσ(L_ν),

where the spectra σ(L_ν) are discrete for each ν∈Ω as L_ν has compact resolvent.

Since u_∞' is a solution to (<ref>), 0 is an eigenvalue of L_0. Now, provided 0 is simple as an eigenvalue of L_0, the implicit function theorem provides a surface λ_0  U →, where U ⊂^2 is a neighborhood of 0, such that λ_0(0) = 0 and λ_0(ν) is a simple eigenvalue of L_ν for any ν∈ U. By symmetry of the spectrum of the real operator  and the decomposition (<ref>), the surface λ_0[Ω] ⊂σ() touches the imaginary axis generically quadratically. So, even in the most stable scenario, where the spectrum of  is bounded away to the left of the imaginary axis except for a quadratic touching at the origin, there is no spectral gap. This leads to the following definition.


The planar wave train u(x,y,t)=u_∞(kx-ω t) to (<ref>) is spectrally stable if there exists η,ϵ > 0 such that


  (D1) L_0 has no spectrum in (λ) ≥ -η besides a simple eigenvalue λ = 0;

  (D2) It holds (σ(L_ν)) ≤ -η |ν|^2 for all ν∈Ω with |ν| ≤ϵ;

  (D3) It holds (σ(L_ν)) < -η for all ν∈Ω with |ν| ≥ϵ.



Throughout this paper we require the spectral stability assumptions (D1)-(D3) to hold true. Similar spectral assumptions are made in one spatial dimension – see <cit.>. We emphasize that spectrally-stable planar wave trains to (<ref>) can be generated from spectrally-stable wave trains to (<ref>) – see <ref>. Examples of such wave trains in one spatial dimension can be found in <cit.>. Note that by Sturm-Liouville theory wave-train solutions to (<ref>) can only be spectrally stable if n > 1, i.e. if (<ref>) is a proper system of reaction-diffusion equations.



 §.§ Statement of results
 

Our results concern nonlinear diffusive stability of spectrally-stable planar wave-train solutions u(x,y,t)=u_∞(kx-ω t) to (<ref>) against classes of spatially localized and spatially nonlocalized perturbations. In our first result, we take a unit vector w∈^2 and consider nonlocalized W^2,∞-perturbations that are bounded, when restricted to the family of lines

    { z∈^2  w· z = a},      a ∈,

but that are exponentially localized in any other spatial direction. We establish diffusive Gaussian-like decay of the perturbation and its derivatives with rate t^-1/2. In addition, the perturbation stays bounded for all times t ≥ 0 on lines of the form (<ref>) and exponentially localized in any other spatial direction. If we account for translational invariance and allow for a phase shift, it is possible to obtain decay with rate log(t)/t. Thus, our result is as follows.

 
Assume (D1)-(D3) hold true and let (β,γ) ∈^2 be a unit vector. There exists constants E_0>0 and C,M > 1 such that for all v_0 ∈ W^2,∞(^2,^n) satisfying

    sup_, y ∈ e^|β + γ y|^2/M(v_0(,y) + ∂_ v_0(,y) + ∂_ v_0(,y)) ≤ E_0,

there exists a solution (t) to (<ref>), with initial condition (0) = u_∞ + v_0, for all time t ≥ 0 satisfying

    (,y,t) - u_∞()≤ Ce^-|β + γ y + αβ t|^2/M(1 + t)/√(1+t),     D_,y^𝔠((,y,t) - u_∞())≤ Ce^-|β + γ y + αβ t|^2/M(1 + t)/√(t),

for ,y ∈ and t > 0, where α is as in Lemma <ref> and 𝔠∈_≥ 0^2 is a multi-index with |𝔠| = 1. In addition, there exists a function ψ_≥ 0→  W^3,∞(^2,) such that

    ( + ψ(,y,t),y,t) - u_∞()   ≤ Clog(2+t)/1+t e^-|β + γ y + αβ t|^2/M(1 + t),
    ψ(,y,t)≤e^-|β + γ y + αβ t|^2/M(1 + t)/√(1+t),        D_,y,t^𝔟ψ(,y,t)≤e^-|β + γ y + αβ t|^2/M(1 + t)/1+t,

for ,y ∈, t ≥ 0 and 𝔟∈_≥ 0^3 with 1 ≤ |𝔟| ≤ 3.


Next, we study the stability of the planar wave-train solution u(x,y,t)=u_∞(kx-ω t) to (<ref>) against exponentially localized perturbations. We obtain diffusive Gaussian-like decay of the perturbation with rate t^-1, which can be improved to t^-3/2 by tracking the phase shift. Moreover, the perturbation stays exponentially localized for all time t ≥ 0. This leads to the following result.

 
Assume (D1)-(D3) hold true. There exists constants E_0>0 and C,M > 1 such that for all v_0 ∈ W^1,∞(^2,^n) satisfying

    sup_, y ∈ e^^2 + y^2/M(v_0(,y) + ∂_ v_0(,y)) ≤ E_0,

there exists a solution (t) to (<ref>), with initial condition (0) = u_∞ + v_0, for all time t ≥ 0 satisfying

    (,y,t) - u_∞()   ≤ Ce^-| + α t|^2 + y^2/M(1 + t)/1+t,     ∂_((,y,t) - u_∞())≤ C e^-| + α t|^2 + y^2/M(1 + t)log(2+t)/√(t)√(1+t),
       ∂_y (,y,t)≤ C e^-| + α t|^2 + y^2/M(1 + t)log(2+t)/√(t)(1+t),

for ,y ∈ and t > 0, where α is as in Lemma <ref> and 𝔠∈_≥ 0^2 is a multi-index with |𝔠| = 1. In addition, there exists a function ψ_≥ 0→  W^3,∞(^2,) such that

    ( + ψ(,y,t),y,t) - u_∞()   ≤ Ce^-| + α t|^2 + y^2/M(1 + t)/(1+t)√(1+t),
    ψ(,y,t)≤ Ce^-| + α t|^2 + y^2/M(1 + t)/1+t,        D_,y,t^𝔟ψ(,y,t)≤ Ce^-| + α t|^2 + y^2/M(1 + t)/(1+t)√(1+t),

for ,y ∈, t ≥ 0 and 𝔟∈_≥ 0^3 with 1 ≤ |𝔟| ≤ 3.


To prove Theorems <ref> and <ref> we decompose the temporal Green's function associated with the parabolic operator ∂_t - and we obtain pointwise bounds on each of the components of the Green's function. These pointwise Green's function estimates are given in Section <ref>. In Section <ref>, we perturb the planar wave-train solution u(,y,t) = u_∞() to (<ref>) and establish nonlinear equations for the perturbation and its derivatives. We track the perturbed solution in two different coordinate systems, where one of these coordinates is used to correct for the phase shift of the solution relative to the original wave train. This leads to a closed nonlinear iteration scheme. We apply the pointwise Green's function bounds to this nonlinear iteration scheme in Section <ref> to prove Theorems <ref> and <ref>.


The pointwise nonlinear stability analysis of wave-train solutions in one spatial dimension presented in <cit.> assumes only that perturbations are algebraically localized. More precisely, perturbations v_0 ∈ H^2(,^n) need to satisfy only that |v_0(x)| ≤ E_0(1+|x|)^-r for x ∈ with 0 < E_0 ≪ 1 and r > 2. The associated perturbed solution then exhibits both algebraic decay and diffusive Gaussian-like decay. By transferring the estimates in <cit.> to our setting, we expect that the exponential weights in our main results can be replaced by algebraic weights. However, since the estimates in <cit.> are technically quite involved and the main point of our analysis is the possibility of nonlocalized perturbations rather than their precise decay properties, we chose to work with exponential weights in this paper for clarity of exposition.




§ GREEN'S FUNCTION DECOMPOSITION AND POINTWISE ESTIMATES
 

In this section, we show that assumptions (D1)-(D3) on the spectrum of the linearization  of (<ref>) about the wave train u(,y,t) = u_∞() lead to a decomposition of the associated temporal Green's function. We obtain pointwise bounds on each of the components of the Green's function by following the (by now seminal) approach, which was introduced in <cit.> and further extended in <cit.> to the setting of wave trains in one spatial dimension. The decomposition of and the pointwise estimates on the Green's function are the starting point of our nonlinear stability analysis, which is performed in the <ref>.



 §.§ Consequences of spectral assumptions
 

We apply the implicit function theorem to expand the spectrum of  about the origin. The obtained control on the spectrum is crucial for the decomposition of the temporal Green's function.
 
Assume (D1). We complexify ν and consider the family L_ν, ν∈^2 of operators on L_^2([0,1],^n) given by (<ref>). There exists a neighborhood U ⊂^2 of 0 and an analytic function λ_0  U → such that the spectrum of L_ν in (λ) ≥ -η consists of the simple eigenvalue λ_0(ν) only. In addition, we have the expansion

    λ_0(ν)    = iαν_x - θν_x^2 - d_⊥ν_y^2 + (ν),     ν = (ν_x,ν_y) ∈ U,

with residual U → satisfying |(ν)| ≤ C|ν|^3 for some constant C>0 and coefficients

    α   = 2k^2⟨ũ_, D  ũ_∞”⟩_2 + ω∈,    d_⊥ = ⟨ũ_, D  ũ_∞'⟩_2 ∈,   θ∈,

where ũ_∞∈(L_0) is the restriction of the 1-periodic wave train u_∞ to [0,1], ũ_ is contained in (L_0^*) such that ⟨ũ_, ũ_∞'⟩_2 = 1 and ⟨·,·⟩_2 denotes the L_^2([0,1],^n)-inner product.

Moreover, for all ν∈ U the kernels of L_ν - λ_0(ν) and its adjoint (L_ν - λ_0(ν))^* are spanned by analytic eigenfunctions q  U → L_^2([0,1],^n) and q_ U → L_^2([0,1],^n), respectively, satisfying ⟨ q(ν), q_(ν)⟩_2 = 1, q(0) = ũ_∞' and q_(0) = ũ_. Finally, the associated derivative maps U → L^2_([0,1],^n) given by ν↦∂_ q(ν) and ν↦∂_ q_(ν) are also analytic.

 Since (D1) is satisfied,  it follows from standard perturbation theory <cit.> that for ν in some neighborhood U ⊂^2 of 0 the spectrum of L_ν in (λ) ≥ -η consists of a simple eigenvalue λ_0(ν) only, which depends analytically on ν. The fact that the associated eigenvectors q(ν) and q_(ν) and their derivatives ∂_ q(ν) and ∂_ q_(ν) are also analytic in ν follows by writing the eigenvalue problem (L_ν - λ) u = 0 as a first order system (∂_ - A(,λ))ϕ = 0 with ϕ = (u,u_) and applying perturbation results from <cit.>.

Since λ_0(ν) is algebraically simple and L_0 is Fredholm of index 0 by (D1), the inner product of q(ν) and q_(ν) cannot vanish by the Fredholm alternative for any ν∈ U. Therefore, we can assume without loss of generality that ⟨ q(ν), q_(ν)⟩_2 = 1 and q(0) = ũ_∞' for any ν∈ U. We define ũ_ = q_(0). The derivative ∂_ν_x q satisfies the equation

    (L_ν - λ_0(ν)) [∂_ν_x q](ν) = [∂_ν_xλ_0](ν) q(ν) - 2k^2 i D (∂_ + i ν_x) q(ν) - ω i q(ν),   ν∈ U.

Taking the inner product with q_(ν) on both sides and evaluating at ν = 0 yields [∂_ν_xλ_0](0) = α i. Similarly, one obtains [∂_ν_yλ_0](0) = 0, [∂_ν_xν_yλ_0](0) = 0 and [∂_ν_yν_yλ_0](0) = -2d_⊥. Since  is a real operator, its spectrum in  is symmetric in the real axis. Therefore, the decomposition (<ref>) yields that θ := -[∂_ν_xν_xλ_0](0)/2 must be real. The expansion (<ref>) now follows by analyticity of λ_0  U →.

The spectral control about the origin obtained in Lemma <ref> reduces the verification of assumption (D2) to checking the signs of two Melnikov-type integrals.

Assume (D1) is satisfied, then assumption (D2) is satisfied if and only if d_⊥,θ > 0.

Notice that the linearization of the reaction-diffusion system (<ref>) in one spatial dimension about the wave train solution u(x,t)=u_∞(kx-ω t) is given by the operator _0 – see (<ref>). Spectral stability of u(x,t) = u_∞(kx-ω t) as a solution to (<ref>), in the sense of <cit.>, entails that 0 is a simple eigenvalue of L_0 and that the spectrum of _0 lies to the left of the imaginary axis, except for a quadratic touching at the origin. This leads to the following result.
 
Assume the wave train u(x,t)=u_∞(kx-ω t) is spectrally stable as solution to (<ref>), in the sense of <cit.>. If it holds d_⊥ > 0, then (D1)-(D2) are satisfied. Moreover, if D = _n, then the planar wave train u(x,y,t)=u_∞(kx-ω t) is spectrally stable as solution to (<ref>).




 §.§ Pointwise Green's function estimates


We follow <cit.> and decompose the temporal Green's function associated with the parabolic operator ∂_t - in a translational mode and a residual. We obtain pointwise estimates for each of these components of the Green's function.

 
Assume (D1)-(D3) hold true. Let χ_≥ 0→ [0,1] be a smooth cut-off function such that χ(t) = 0 for 0 ≤ t ≤ 1 and χ(t) = 1 for t ≥ 2. The temporal Green's function G(,,y,t) associated with the operator ∂_t - in (<ref>) can be decomposed as

    G(,,y,t)    = u_∞'()e(,,y,t) + G(,,y,t),
    
    e(,,y,t)    = χ(t)/4 π t√(d_⊥θ)e^-|-+α t|^2/4θ t - |y|^2/4d_⊥ t u_ ()^*,
      ,,y ∈, t ≥ 0,

where α,θ and d_⊥ are as in Lemma <ref> and u_→^n is the periodic extension of the eigenfunction ũ_∈ L^2_([0,1],^n) of the adjoint L_0^* (see also Lemma <ref>). For each j ∈_≥ 0 and multi-indices 𝔞, 𝔟∈_≥ 0^3 with |𝔞| = 1 and |𝔟| ≥ 0, there exists C,M > 1 such that we have the pointwise estimates

    (,,y,t)   ≤ Ct^-1(1+t)^-1/2e^-|- + α t|^2 + |y|^2/Mt,
    D_,,y^𝔞(,,y,t)   ≤ Ct^-3/2e^-|- + α t|^2 + |y|^2/Mt,
    D_,y,t^𝔟∂_^j e(,,y,t)   ≤ C(1+t)^-1-|𝔟|/2e^-|- + α t|^2 + |y|^2/Mt,
        ,,y ∈, t > 0.



To prove Theorem <ref> we employ similar methods as in <cit.>, where pointwise Green's function estimates are obtained for wave-train solutions to reaction-diffusion systems in one spatial dimension. To account for an additional spatial direction, we combine the methods in <cit.> with those from <cit.>, where pointwise bounds are established for spatially multidimensional viscous shock fronts. Since the proof of Theorem <ref> follows, by and large, the nontrivial, but by now classical, approach introduced in <cit.>, we decided to include the proof of Theorem <ref> in Appendix <ref>.



§ NONLINEAR ITERATION SCHEME
 

In this section, we perturb the planar wave-train solution u(,y,t) = u_∞() to (<ref>) and we establish nonlinear equations for the perturbation and its derivatives. To account for translational invariance and exploit the decomposition (<ref>) of the Green's function, we introduce a phase function that tracks the shift of the perturbed solution in space relative to the original wave train. Our goal is to obtain a closed nonlinear iteration scheme for the perturbation and the phase function, which will be employed in <ref> to prove the nonlinear stability results in <ref>.



 §.§ Perturbation equations

We consider the perturbed solution

    (,y,t) = u_∞() + (,y,t),

to (<ref>). The perturbation  and its -derivatives satisfy

    (∂_t - )[∂_^i ] = _i,      i = 0,1,2,

with

    _0    := f(u_∞+) - f(u_∞) - f'(u_∞) , 
    _1    := (f'(u_∞+)-f'(u_∞))(u_∞' + _),
    _2    := f”(u_∞ + )(u_∞' + _) _ + (f”(u_∞ + )(u_∞' + _) - f”(u_∞)u_∞')u_∞'
             + (f'(u_∞+)-f'(u_∞))(u_∞” + _),

where we suppress the argument (,y,t) of  and its derivatives, and the argument  of u_∞ and its derivatives for notational convenience. Using Taylor's Theorem, one observes that _0 is quadratic in , whereas _1 and _2 contain linear terms in  or its derivatives. More precisely, there exist constants B,C > 1 such that as long as (t)_∞≤ B, we have the bounds

    _0(,y,t)≤    C(,y,t)^2,     _1(,y,t)≤ C(,y,t)(1+_(,y,t)),

and, as long as (t)_∞,_(t)_∞≤ B, we have the bound

    _2(,y,t)   ≤ C((,y,t)(1+_(,y,t)) + _(,y,t)),

for ,y ∈ and t ≥ 0, where ·_∞ denotes the L^∞(^2,^n)-norm. To account for translational invariance – see Remarks <ref> and <ref> – we introduce another coordinatization for the perturbed solution  to (<ref>). As in <cit.>, we write

    ( + ψ(,y,t),y,t) = u_∞() + v(,y,t),

where the phase function ψ^2 ×_≥ 0→ is to be determined later. By the mean value theorem, there exist constants B,C > 1 such that as long as ψ_(t)_∞≤ B, we have the bounds

    ∂_^i (v - )(,y,t)≤    C∑_j = 0^i(∂_^j+1 u_∞_∞ + ∂_^j+1(t)_∞) ∂_^i-jψ(,y,t), 
    (v_y - _y)(,y,t)≤    _ y(t)_∞ψ(,y,t) + (u_∞'_∞ + _(t)_∞)ψ_y(,y,t),
    (v_ y - _ y)(,y,t)≤    C(_ y(t)_∞ψ(,y,t) + _ y(t)_∞ψ_(,y,t).
            + (u_∞”_∞ + _(t)_∞) ψ_y(,y,t)
           .+ (u_∞'_∞ + _(t)_∞) ψ_ y(,y,t)),

for ,y ∈, t ≥ 0 and i = 0,1,2. Our next step is to derive an equation for the perturbation v, the phase ψ and their derivatives. We introduce

    w(,y,t) := ( + ψ(,y,t),y,t) = u_∞() + v(,y,t).

Substituting w(,y,t) into (<ref>), while using that (,y,t) solves (<ref>), leads to the residue induced by the phase

    w_t - D(k^2 w_ + w_yy)    - ω w_ - f(w) =
       -_3ψ_ + _1ψ_t + D(_22ψ_ - _1 2ψ_y - (_1 ψ_y)_y - k^2 (_1 ψ_)_) + f()ψ_,

which yields

    (∂_t - )v =    f(u_∞ + v) - f(u_∞) - f'(u_∞)v
       - _3ψ_ + _1ψ_t + D(_22ψ_ - _1 2ψ_y - (_1 ψ_y)_y - k^2 (_1 ψ_)_) + f()ψ_,

where we suppress the arguments (,y,t) of v,w,ψ and their derivatives, the arguments (+ψ(,y,t),y,t) of  and its derivatives and the arguments  of u_∞ and its derivatives. To avoid confusion by suppressing the arguments of , we denote the partial derivative of  with respect to its j-th argument by _j. In order to obtain a closed system in v and ψ, we need to eliminate the -terms from the right-hand side of (<ref>). Therefore, we express _1, _2 and _3 in terms of v, ψ and their derivatives as follows. First, we calculate

    u_∞' + v_ = w_ = _1(1+ψ_),    v_t = w_t = _3 + _1ψ_t,    v_y = w_y = _2 + _1ψ_y,

yielding the identities

    _1    = 1/1+ψ_(u_∞' + v_)   _2 = v_t - ψ_t/1+ψ_(u_∞' + v_),   _3 = v_y - ψ_y/1+ψ_(u_∞' + v_),

where we suppress the arguments again. By employing the identities

    (∂_t - )[u_∞' ψ] =    [u_∞'] ψ - D(k^2 (u_∞' ψ_)_ + u_∞' ψ_yy) - (D k^2 u_∞” + ω u_∞') ψ_ + u_∞' ψ_t
    
    =    - D(k^2 (u_∞' ψ_)_ + u_∞' ψ_yy) + f(u_∞)ψ_ + u_∞' ψ_t,
    (∂_t - )[vψ_] =    v_t ψ_ + vψ_ t - D(k^2(v_ψ_ + 2v_ψ_ + v ψ_) + v_yyψ_ + 2v_yψ_ y + v ψ_ yy)
            - ω(v_ψ_ + vψ_) - f'(u_∞) v ψ_,

and substituting (<ref>) into the right hand side of (<ref>), we obtain the perturbation equation

    (∂_t - )[v - u_∞' ψ] =  - (∂_t - )[vψ_],

where the nonlinearity  is given by

    :=   (f(u_∞+v) - f(u_∞) - f'(u_∞) v)(1+ψ_) + v_ψ_t + vψ_ t - ω(v_ψ_ + v ψ_)
            - D(v(k^2ψ_ + ψ_ yy) + v_(3k^2 ψ_ + ψ_yy) + 2v_yψ_ y + 2k^2 v_ψ_ + 2v_ yψ_y)
            + D/1+ψ_(2(u_∞' + v_)(ψ_yψ_ y + k^2 ψ_ψ_) + (u_∞” + v_)(ψ_y^2 + k^2 ψ_^2))
            - D/(1+ψ_)^2(u_∞' + v_)(ψ_y^2 + k^2 ψ_^2)ψ_.

Using Taylor's Theorem, it is relatively straightforward to observe that the nonlinearity (<ref>) is quadratic in v, ψ and their derivatives. More precisely, there exist constants B,C > 1 such that as long as it holds

    v(t)_∞ + ∑_|𝔞| = 1D_,y^𝔞ψ(t)_∞≤ B,

we have the bound

    (,y,t)≤    C[(v(,y,t) + ∑_1 ≤ |𝔞| ≤ 3D_,y,t^𝔞ψ(,y,t))^2.
       .     + (v_y(,y,t) + ∑_0 ≤ |𝔟| ≤ 1D_,y^𝔟 v_(,y,t)) ∑_1 ≤ |𝔠| ≤ 2D_,y,t^𝔠ψ(,y,t)],

for ,y ∈ and t ≥ 0.

 
In the perturbation equation (<ref>) for v, we grouped terms that are nonlinear in (v,ψ) and their derivatives on the right-hand side, whereas the left-hand side contains all contributions that are linear in (v,ψ) and their derivatives. This decomposition suggests new coordinates (w,ψ) with w = v + u_∞'ψ. The advantage of working in these coordinates is that, by choosing ψ appropriately, the contribution u_∞' ψ accounts for the translational mode u_∞'()e(,,y,t) of the Green's function in the nonlinear iteration – see Remark <ref>. The principle of canceling the slowly decaying translational mode via these type of coordinatizations has been developed independently in <cit.> and <cit.> using different methods – see <cit.> for a review.

In the nonlinear right-hand side of (<ref>), we introduced the term (∂_t - )[vψ_] in order to eliminate any temporal derivatives of v in the nonlinear iteration. Indeed, one observes from (<ref>) that the residual nonlinearity  contains only - and y-derivatives of v. In Remark <ref> below, we will explain how we control spatial derivatives of v in the nonlinear iteration.




 §.§ Duhamel's integral formulation

Our goal is obtain a closed nonlinear iteration scheme by integrating (<ref>), while exploiting the decomposition (<ref>) of the Green's function. Applying Duhamel's formula to (<ref>) yields the equivalent integral equation

    v(,y,t) =    u_∞'()ψ(,y,t) - v(,y,t)ψ_(,y,t) + ∫_^2 G(,,y-,t) v_0(,) d d
            + ∫_0^t ∫_^2 G(,,y-,t-s) (,,s)d d ds,

with ,y ∈, t ≥ 0 and v_0(,y) = v(,y,0). Now we define the phase function ψ as the solution to the integral equation

    ψ(,y,t) =     -∫_^2 e(,,y-,t) v_0(,) d d
            - ∫_0^t ∫_^2 e(,,y-,t-s) (,,s)d d ds.

Since it holds e(,,y,0) = 0 for all ,,y ∈ by construction, we have thus taken ψ(,y,0) = 0 as initial condition for ψ. This choice of ψ leads via (<ref>) to the system of integral equations

    D_,y^𝔞 v(,y,t) =    -D_,y^𝔞 v(,y,t) ψ_(,y,t) + ∫_^2 D_,y^𝔞(,,y-,t) v_0(,) d d
            + ∫_0^t ∫_^2 D_,y^𝔞(,,y-,t-s) (,,s)d d ds,
    
    D_,y,t^𝔟ψ(,y,t) =    -∫_^2 D_,y,t^𝔟 e(,,y-,t) v_0(,) d d
           - ∫_0^t ∫_^2 D_,y,t^𝔟 e(,,y-,t-s) (,,s)d d ds,

with ,y ∈, t ≥ 0, and multi-indices 𝔞∈_≥ 0^2 and 𝔟∈_≥ 0^3 satisfying 0 ≤ |𝔞|≤ 1 and 0 ≤ |𝔟| ≤ 3, where we use that e(,,y,0) = 0 for all ,,y ∈ to determine the t-derivative of the second integral in (<ref>).

To obtain a closed nonlinear iteration scheme from (<ref>), we still need to control the second derivatives v_ and v_ y in the nonlinearity (<ref>). We emphasize that we cannot obtain such control by differentiating the v_- and v_y-equation in (<ref>) with respect to , since, as explained in the subsequent Remark <ref>, the second spatial derivatives of the Green's function (,,y,t) do not permit t-integrable bounds. Instead, we proceed as follows. First, we eliminate the second derivatives v_ and v_ y in the v-equation in (<ref>) using integration by parts at the expense of taking more derivatives of the ψ-dependent terms. We note that this elimination is only possible due to the fact that all derivatives of v in (<ref>) are paired with ψ-terms and not with other v-terms, and due to the fact that the pointwise bounds on the Green's functions ∂_(,,y,t) and ∂_y(,,y,t), obtained in Theorem <ref>, are integrable over t. Since we control also higher order - and y-derivatives of the Green's function e(,,y,t) via Theorem <ref>, we can apply integration by parts multiple times in the ψ-equation to eliminate all derivatives of v from the nonlinearity. The elimination of (second) derivatives of v from the nonlinearities in the v- and ψ-equations in (<ref>) lead to sharp estimates, which are necessary to close the nonlinear iteration. However, the second derivatives of v cannot be eliminated from the nonlinearity in the v_- and v_y-equations in (<ref>) using integration by parts, since this would again lead to second spatial derivatives of the Green's function (,,y,t), which do not permit t-integrable bounds. To control the second derivatives v_ and v_ y of v, we will append integral equations for the derivatives _,_ y, _ and _ y of the `unshifted' perturbation  to the scheme.

Thus, exploiting that (,,y,t), e(,,y,t) and its derivatives are exponentially localized in space and assuming that there exists a bound B > 1 such that

    sup_0 ≤ s ≤ t(v(s)_∞ + v_(s)_∞ + ∑_1 ≤ |𝔟| ≤ 3D_,y^𝔟ψ(s)_∞) ≤ B,

we integrate by parts in (<ref>) to obtain the following equivalent integral system

    v(,y,t) =    -v(,y,t) ψ_(,y,t) + ∫_^2(,,y-,t) v_0(,) d d
            + ∫_0^t ∫_^2(,,y-,t-s) _1(,,s)d d ds 
            + ∫_0^t ∫_^2∂_(,,y-,t-s) _2(,,s)d d ds
            + ∫_0^t ∫_^2∂_y (,,y-,t-s) _3(,,s)d d ds
    
    D_,y^𝔠 v(,y,t) =    -D_,y^𝔠[v(,y,t) ψ_(,y,t)] + ∫_^2 D_,y^𝔠(,,y-,t) v_0(,) d d
            + ∫_0^t ∫_^2 D_,y^𝔠(,,y-,t-s) (,,s)d d ds,
    
    D_,y,t^𝔟ψ(,y,t) =    -∫_^2 D_,y,t^𝔟 e(,,y-,t) v_0(,) d d
            + ∑_0 ≤ |𝔞| ≤ 2∫_0^t ∫_^2 D_,y,t^𝔟 D_,y^𝔞 e(,,y-,t-s) _𝔞(,,s)d d ds,

with ,y ∈, t ≥ 0 and multi-indices 𝔠∈_≥ 0^2 and 𝔟∈_≥ 0^3 satisfying |𝔠|=1 and 0 ≤ |𝔟| ≤ 3, where the nonlinearity _1 contains the term

    Dv_(2ψ_y y + (2k^2ψ_ - ψ_y^2 + k^2ψ_^2/1+ψ_)_),

with a derivative of v, whereas all other terms present in _1 do not depend on derivatives of v. Similarly, the nonlinearity _2 only contains the term

    D v_(2k^2ψ_ - ψ_y^2 + k^2ψ_^2/1+ψ_),

with a derivative of v, the nonlinearity _3 only contains the term

    -2D v_ψ_y,

with a derivative of v and the nonlinearities _𝔞 contain no derivatives of v for all multi-indices 𝔞∈_≥ 0^2 with 0 ≤ |𝔞| ≤ 2. Using Taylor's Theorem, one readily observes that the obtained nonlinearities _j,_𝔞 are quadratic in v, ψ and their derivatives. More precisely, there exist constants B,C > 1 such that as long as

    v(t)_∞ + ∑_1 ≤ |𝔟| ≤ 2D_,y^𝔟ψ(t)_∞≤ B,

we have the bounds

    _j(,y,t)≤    C[(v(,y,t) + ∑_1 ≤ |𝔟| ≤ 3D_,y,t^𝔟ψ(,y,t))^2. 
       .                         + v_(,y,t)∑_1 ≤ |𝔠| ≤ 2D_,y^𝔠ψ(,y,t)],
    _𝔞(,y,t)≤    C(v(,y,t) + ∑_1 ≤ |𝔟| ≤ 3D_,y,t^𝔟ψ(,y,t))^2,

for ,y ∈, t ≥ 0, j = 1,2,3 and 𝔞∈_≥ 0^2 with 0 ≤ |𝔞| ≤ 2. Comparing the estimates on _j, j = 1,2,3 to the estimate (<ref>) on , we observe that we have indeed gained one derivative of v. Since the bounds on the first spatial derivatives of v, obtained in the subsequent analysis in <ref>, exhibit stronger decay than those on the second spatial derivatives of v, this will lead to sharp estimates on v, and on ψ and its derivatives.

However, as mentioned before, the integral equations (<ref>) do not give rise to a closed nonlinear iteration scheme yet, because we still need to control the second derivatives v_ and v_ y occurring in the nonlinearity  in the v_- and v_y-equations in (<ref>). This can be achieved by appending integral equations for the derivatives _,_ y, _ and _ y of the `unshifted' perturbation  to the scheme (<ref>) and employing the estimates (<ref>) – see also Remark <ref>. Applying Duhamel's formula to (<ref>) yields

    D_,y^𝔞∂_^i (,y,t) =    ∫_^2 D_,y^𝔞 G(,,y-,t) ∂_^i v_0(,) d d
            + ∫_0^t ∫_^2 D_,y^𝔞 G(,,y-,t-s) _i(,,s)d d ds,

with ,y ∈, t ≥ 0, i = 0,1,2 and 𝔞∈_≥ 0^2 satisfying 0 ≤ |𝔞| ≤ 1, where we used that ψ(,y,0) = 0 implies v_0(,y) = v(,y,0) = (,y,0) for all , y ∈. Observe that by (<ref>) and (<ref>) the nonlinearities _i, i = 0,1,2, can be estimated in terms of , _ and _. Thus, combining the integral equations (<ref>) and (<ref>) and the estimates (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) yields a closed nonlinear iteration scheme.


Applying Duhamel's formula to (<ref>) gives the integral equation

    (,y,t) =    ∫_^2 G(,,y-,t) v_0(,) d d
            + ∫_0^t ∫_^2 G(,,y-,t-s) _0(,,s)d d ds,

with ,y ∈ and t ≥ 0. Observe that (<ref>) yields a closed nonlinear iteration scheme, since the nonlinearity _0 depends only on  itself. However, because _0 is quadratic in  in general, the Green's function estimate

    G(,,y,t)≤ Ct^-1e^-|- + α t|^2 + |y|^2/Mt,     ,,y ∈, t > 0,

obtained in Theorem <ref> is not strong enough to close a nonlinear iteration argument. The decomposition in Theorem <ref> factors out the translational mode u_∞'()e(,,y,t) of the Green's function, which yields an additional decay factor t^-1/2 in the estimate of the residual. Moreover, the introduction of the phase function ψ(,y,t) in the nonlinear iteration scheme leads to a contribution u_∞'ψ in (<ref>), which cancels the translational mode of the Green's function by a judicious choice (<ref>) of ψ. The additional decay of the residual mode of the Green's function can then be exploited to close the nonlinear iteration argument. The trade-off is that the introduction of the phase function ψ(,y,t) yields derivatives of v in the nonlinearity, which needs to be controlled – see also Remark <ref>.


 
In order to obtain a closed nonlinear iteration scheme from (<ref>), one needs to control the second derivatives v_ y and v_ in the nonlinearity (<ref>). Naively, one might introduce integral equations for these second derivatives of v by differentiating (<ref>). However, the second order derivatives of the Green's function satisfy the bounds

    D_,,y^𝔟(,,y,t)≤ Ct^-3/2(1+t^-1/2)e^-|- + α t|^2 + |y|^2/Mt,     ,,y ∈, t > 0, 𝔟∈_≥ 0^3, |𝔟| = 2,

for some constants C,M > 1, which leads, after integration over  and y, to the nonintegrable factor (t-s)^-1 in the estimation of the integral equations for v_ y and v_. Integrating by parts to move a derivative from the Green's function to the nonlinearity, introduces third derivatives of v in the nonlinearity and thus transfers the problem of controlling second derivatives of v to controlling third derivatives of v.

The same problem occurs in the nonlinear stability analyses <cit.> of wave-train solutions to reaction-diffusion systems in one spatial dimension. There one establishes a nonlinear damping estimate that controls the H^k-norm of the perturbation v for k ≥ 0 by its L^2-norm and by the H^k-norm of (ψ_,ψ_t). This yields nonlinear L^1 ∩ H^k → H^k-stability in <cit.> and nonlinear L^1 ∩ H^k → L^∞-stability with nonlocal phase in <cit.>. The damping estimate from <cit.> is also employed in the pointwise nonlinear stability analyses in <cit.>. Since the derivatives of v are paired with ψ-terms in the nonlinearity, one only needs L^∞-bounds on the derivatives of v in <cit.>. These L^∞-bounds follow via Sobolev interpolation from  H^k-bounds on v that are obtained using the damping estimate from <cit.>.

One can establish a 2-dimensional equivalent of the damping estimate in <cit.>, which bounds the H^k(^2,^n)-norm of v in terms of its L^2(^2,^n)-norm and the H^k(^2,^n)-norm of (ψ_,ψ_t). However, such a damping estimate cannot be employed in our analysis, since we allow for nonlocal perturbations. More precisely, we consider planar perturbations that are not in L^2(^2,^n), whereas in <cit.> the perturbations are L^2-localized on the real line (allowing possibly for a nonlocal phase modulation). Since we consider perturbations that are only nonlocalized in one spatial direction (say the y-direction), it may be possible to adapt the damping estimate in <cit.> to work in the mixed L^𝔭-space L^𝔭(^2,^n), where 𝔭 is the tuple (2,∞), endowed with the norm

    u_𝔭 = sup_y ∈(∫_u(,y)^2 d)^1/2.

However, we expect that one needs to append a damping estimate of v_y in L^2(^2,^n) in order to control the y-derivatives in the damping estimate of v in L^𝔭(^2,^n), thus requiring the derivatives of v to be localized in ^2.

In this paper, we choose an alternative approach by appending the integral equations (<ref>) for the derivatives of the `unshifted' perturbation  to the nonlinear iteration scheme (<ref>). All derivatives in the nonlinearity (<ref>) vanish, when the phase function ψ is set to 0. Therefore, the spatial derivatives of  can be controlled without running into problems due to nonintegrable Green's function bounds. Then, estimates like (<ref>) can be used to control the derivatives of v. The advantage over a possible L^𝔭(^2,^n)-damping estimate is that we do not require localization of any derivative of v in ^2. The trade-off is that the obtained estimates on v_ and v_ y are not as strong as one might expect due to the fact that we do not correct for a phase shift relative to the original wave train while estimating  and its derivatives. However, by combining these estimates with the ones on the integral scheme (<ref>) which does track the phase shift, they are strong enough to close the nonlinear iteration argument – see Remark <ref>. Moreover, for fully exponentially localized perturbations, our method yields the same optimal decay bounds as the ones one obtains with a damping estimate in H^k(^2,^n) up to a possibly artificial log(2+t)-factor – see Theorem <ref> and Remark <ref>.




§ NONLINEAR STABILITY ANALYSIS
 

We prove the nonlinear stability results stated in <ref> by applying the pointwise Green's function estimates from Theorem <ref> to the nonlinear iteration scheme consisting of (<ref>), (<ref>) and (<ref>). We start with the proof of Theorem <ref> concerning nonlocalized perturbations.

 In this proof, the constant M > 1 is as in Theorem <ref>, whereas C>1 denotes a constant, which is independent of E_0,,,y and t, that will be taken larger if necessary.

Short-time existence theory for semilinear parabolic equations – see <cit.> – yields via a standard contraction-mapping argument that there exists a maximal T_0 ∈ (0,∞] such that (<ref>) has a mild solution (t) in W^2,∞(^2,^n) on [0,T_0) with (0) = v_0. The map t ↦(t)_W^2,∞(^2,^n) is continuous on [0,T_0) and, if T_0 is finite, it must blow up as t ↑ T_0. By a similar contraction-mapping argument, the semilinear parabolic system given by (<ref>), (<ref>) and (<ref>) – and hence the integral system given by (<ref>) and (<ref>) – has a solution (v(t),ψ(t)) in X_∞ := W^2,∞(^2,^n) × W^3,∞(^2,) on some maximal time interval [0,T_1), T_1 ∈ (0,∞] with initial condition (v(0),ψ(0)) = (v_0,0) such that the template function

    η(t)    := sup_[ 0 ≤ s ≤ t;      ,y ∈ ](e^|β + γ y + αβ s|^2/M(1 + s)[(1+s)(v(,y,s)/log(2+s) + ∑_1 ≤ |𝔟| ≤ 3D_,y,s^𝔟ψ(,y,s))..
                   ..∑_1 ≤ |𝔟| ≤ 3 + √(s)(v_(,y,s)+ v_y(,y,s) + _y(,y,s))..
                   ..∑_1 ≤ |𝔟| ≤ 3 + √(1+s)((,y,s) + _(,y,s) +ψ(,y,s))].
           .∑_1 ≤ |𝔟| ≤ 3 +
    √(s)/1+√(s)(v_(,y,s) + v_ y(,y,s) + _(,y,s) + _ y(,y,s))),

is well-defined and continuous on [0,T_1) and, if T_1 is finite, then η(t) must blow up as t ↑ T_1. Our goal is to prove that there exist constants B > 0 and C > 1 such that for all t ≥ 0 with η(t) ≤ B we have

    η(t) ≤ C(E_0 + η(t)^2).

Since η must be continuous as long as it remains small, we can apply continuous induction using estimate (<ref>). Thus, provided that E_0 < 1/4C, it follows η(t) ≤ 2CE_0 for all t ≥ 0, which readily yields the result.

We prove the key estimate (<ref>). We employ two integral identities

    ∫_^2 e^-|- + α t|^2 + |y-|^2/Mt - |β + γ|^2/Md d   = Mπ t e^-|β + γ y + αβ t|^2/M(1 + t)/√(1+t),
    ∫_^2 e^-|- + α (t-s)|^2 + |y-|^2/M(t-s) - |β + γ + αβ s|^2/M(1+s)d d   = Mπ (t-s) √(1+s) e^-|β + γ y + αβ t|^2/M(1 + t)/√(1+t),

with ,y,s,t ∈ satisfying 0 ≤ s < t. These identities are obtained by adding the fractions in the exponent of the integrand, while recalling that (β,γ) ∈^2 is a unit vector, and applying the (standard) formula

    ∫_ e^az(z-2b)dz    = √(π)/√(|a|)e^-ab^2,       a < 0, b ∈,

twice to evaluate the double integral.

Let B > 1 be as in <ref> and assume t > 0 is such that η(t) ≤ B. We start by bounding the solutions to the integral system (<ref>) in terms of η(t). By estimate (<ref>) the nonlinearities in (<ref>) can be bounded as

    _j(,,s)   ≤ Cη(t)^2 e^-|β + γ + αβ s|^2/M(1+s)/(1+s)√(s),
            (,,s)≤ Cη(t)^2 e^-|β + γ + αβ s|^2/M(1+s)/√(1+s)√(s),
       _𝔞(,,s)≤ Cη(t)^2 (log(2+s))^2 e^-|β + γ + αβ s|^2/M(1+s)/(1+s)^2,

with ,∈, 0 < s ≤ t, j = 1,2,3 and 𝔞∈_≥ 0^3 satisfying 0 ≤ |𝔞| ≤ 2 . Thus, applying the pointwise Green's function estimates in Theorem <ref> to the integral system (<ref>) and employing (<ref>) and (<ref>), yields

    v(,y,t)   ≤ C e^-|β + γ y + αβ t|^2/M(1 + t)/√(1+t)(η(t)^2log(2+t)/(1+t)√(1+t) + E_0/√(1+t) +  ∫_0^t η(t)^2/√(1+s)√(s)√(1+t-s) ds)
       ≤ C(E_0+η(t)^2)log(2+t)/1+t e^-|β + γ y + αβ t|^2/M(1 + t),
    D_,y^𝔠 v(,y,t)   ≤ C e^-|β + γ y + αβ t|^2/M(1 + t)/√(t)(η(t)^2/1+t + E_0/√(1+t) + η(t)^2∫_0^t 1/√(s)√(t-s) ds)
       ≤ C(E_0+η(t)^2)e^-|β + γ y + αβ t|^2/M(1 + t)/√(t),
    ψ(,y,t)   ≤ C e^-|β + γ y + αβ t|^2/M(1 + t)/√(1+t)(E_0 + η(t)^2 ∫_0^t (log(2+s))^2/(1+s)√(1+s) ds)
       ≤ C(E_0 + η(t)^2)e^-|β + γ y + αβ t|^2/M(1 + t)/√(1+t),
    D_,y,t^𝔟ψ(,y,t)   ≤ C e^-|β + γ y + αβ t|^2/M(1 + t)/√(1+t)(E_0/√(1+t) + η(t)^2 ∫_0^t (log(2+s))^2/(1+s)√(1+s)√(1 + t - s) ds) 
       ≤ C(E_0 + η(t)^2)e^-|β + γ y + αβ t|^2/M(1 + t)/1+t,

with ,y ∈, and multi-indices 𝔠∈_≥ 0^2 and 𝔟∈_≥ 0^3 satisfying |𝔠| = 1 and 0 ≤ |𝔟| ≤ 3.

Next, we use (<ref>) to bound (,y,t), _(,y,t) and _y(,y,t) in terms of η(t). 
Estimates (<ref>) and (<ref>), and η(t)≤ B yield

    (,y,t)≤    v(,y,t) + (u_∞'_∞ + _(t)_∞) ψ(,y,t)≤ C(E_0 + η(t)^2)e^-|β + γ y + αβ t|^2/M(1 + t)/√(1+t),
    _(,y,t)≤    v_(,y,t) + (u_∞”_∞ + _(t)_∞)ψ(,y,t)
            + (u_∞'_∞ + _(t)_∞) ψ_(,y,t)≤ C(E_0+η(t)^2)e^-|β + γ y + αβ t|^2/M(1 + t)/√(t),
    _y(,y,t)≤    v_y(,y,t) + _ y(t)_∞ψ(,y,t) +  (u_∞'_∞ + _(t)_∞) ψ_y(,y,t)
    ≤    C(E_0+η(t)^2)e^-|β + γ y + αβ t|^2/M(1 + t)/√(t),

with ,y ∈.

Subsequently, we bound solutions to the integral system (<ref>). First, we establish a short-time bound on _. As η(t) ≤ B, estimates (<ref>) and (<ref>) yield

    _1(,,s)   ≤ C(E_0 + η(s)^2) e^-|β + γ + αβ s|^2/M(1+s)/√(1+s),

with ,∈ and 0 ≤ s ≤ t. On the other hand, the pointwise Green's function estimates in Theorem <ref> imply

    G(,,y,s)   ≤ Ce^-|- + α s|^2 + |y|^2/Ms/s,
            ,,y ∈, s > 0.

Therefore, applying (<ref>) and (<ref>) to the integral equation for _ in (<ref>) and employing (<ref>), we obtain

    _(,y,t)   ≤ C e^-|β + γ y + αβ t|^2/M(1 + t)(E_0/√(1+t) + (E_0 + η(t)^2) ∫_0^t 1/√(1+t) ds)
       ≤ C(E_0+η(t)^2)(1+√(t))e^-|β + γ y + αβ t|^2/M(1 + t),

with ,y ∈. Thus, combining (<ref>) (for short time) with (<ref>) (for large time), we arrive at

    _(,y,t)   ≤ C(E_0+η(t)^2)e^-|β + γ y + αβ t|^2/M(1 + t)/√(1+t),     ,y∈.

Second, we bound the third derivatives _ and _ y. As η(t) ≤ B, estimates (<ref>), (<ref>) and (<ref>) imply

    _2(,,s)≤ C(E_0 + η(s)^2)(1+1/√(s)) e^-|β + γ + αβ s|^2/M(1+s)/√(1+s),

with ,∈ and 0 < s ≤ t. Moreover, Theorem <ref> yields

    ∂_ G(,,y,s)   ≤ C(1+1/√(s))e^-|- + α s|^2 + |y|^2/Ms/s,
    ∂_y G(,,y,s)   ≤ Ce^-|- + α s|^2 + |y|^2/Ms/s√(s),
            ,,y ∈, s > 0.

Therefore, applying (<ref>) and (<ref>) to the integral system (<ref>) and employing (<ref>), we obtain

    D_,y^𝔞_(,y,t)   ≤ C e^-|β + γ y + αβ t|^2/M(1 + t)/√(1+t)(E_0(1+1/√(t)) + (E_0 + η(t)^2)∫_0^t (1 +1/√(s)) (1+1/√(t-s))ds) 
       ≤ C(√(t) + 1/√(t))(E_0+η(t)^2),

with ,y ∈ and 𝔞∈_≥ 0^2 satisfying |𝔞| = 1. Combining the interpolation inequality

    u'_∞^2 ≤ Cu_∞u”_∞,      u ∈ W^2,∞(,),

with the estimates on _, _, _y and _ y, established in (<ref>) and (<ref>), we arrive at

    _(t)_∞, _ y(t)_∞≤ C(E_0+η(t)^2)1 + √(t)/√(t).


Now, combining estimates (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) yields

    v_(t)_∞≤    _(t)_∞ + C∑_j = 0^2 (∂_^j+1 u_∞(t)_∞ + ∂_^j+1(t)_∞)∂_^2-jψ(t)_∞
    ≤    C(E_0+η(t)^2)1+√(t)/√(t),
    v_ y(t)_∞≤    _ y(t)_∞ + C(_ y(t)_∞ψ(t)_∞ + _ y(t)_∞ψ_(t)_∞.
           .+ (∂_ u_∞_∞ + _(t)_∞) ψ_y(t)_∞ + (∂_ u_∞_∞ + _(t)_∞) ψ_ y(t)_∞)
    
    ≤    C(E_0+η(t)^2)1+√(t)/√(t).

Finally, estimate (<ref>) follows from (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), which concludes the proof.


The proof of Theorem <ref> concerning exponentially localized perturbations has a similar set-up as the proof of Theorem <ref>. However, the obtained decay estimates differ.

 In this proof, the constant M > 1 is as in Theorem <ref>, whereas C>1 denotes a constant, which is independent of E_0,,,y and t, that will be taken larger if necessary.

Similar to the proof Theorem <ref>, we conclude that the template function

    η(t) :=   sup_[ 0 ≤ s ≤ t;      ,y ∈ ] e^| + α s|^2 + y^2/M(1 + s)[(1+s)√(1+s)(v(,y,s) + ∑_1 ≤ |𝔟| ≤ 3D_,y,s^𝔟ψ(,y,s)) .
         .∑_1 ≤ |𝔟| ≤ 3 + (1+s)((,y,s) + ψ(,y,s)) + √(s)√(1+s)/log(2+s)(_(,y,s) + v_(,y,s))],

is well-defined and continuous on some maximal interval [0,T) with T ∈ (0,∞] and, if T is finite, it must blow up as t ↑ T. Again we prove that there exists constants B > 0 and C > 1 such that for all t ≥ 0 with η(t) ≤ B we have

    η(t) ≤ C(E_0 + η(t)^2).

Then, provided E_0 < 1/4C, we have η(t) ≤ 2CE_0 for all t ≥ 0 by continuous induction, which yields the result.

Thus, we establish the key estimate (<ref>) proceeding as in the proof Theorem <ref>. This time, we consider the closed integral scheme, which consists of the v- and ψ-equations in (<ref>), the _- and _-equations in (<ref>) and the estimates (<ref>), (<ref>), (<ref>) and (<ref>). Moreover, we employ two simpler integral identities

    ∫_^2 e^-|- + α t|^2 + |y-|^2/Mt - ^2 + ^2/Md d   = Mπ t e^-| + α t|^2 + y^2/M(1 + t)/1+t,
    ∫_^2 e^-|- + α (t-s)|^2 + |y-|^2/M(t-s) - | + α s|^2 + ^2/M(1+s)d d   = Mπ (t-s) (1+s) e^-| + α t|^2 + y^2/M(1 + t)/1+t,

with ,y,s,t ∈ satisfying 0 ≤ s < t. These identities are obtained by factorizing the double integral into a product of an integral over  and one over , each of which can be evaluated using (<ref>).

Let B > 1 be as in <ref> and assume t > 0 is such that η(t) ≤ B. As in the proof of Theorem <ref>, we bound the solutions to the integral systems (<ref>) and (<ref>) in terms of η(t).  Thus, by applying Theorem <ref>, (<ref>) and (<ref>) to the integral system (<ref>), we obtain

    v(,y,t)   ≤ C e^-| + α t|^2 + y^2/M(1 + t)/1+t(η(t)^2/(1+t)^2 + E_0/√(1+t) + η(t)^2 ∫_0^t log(2+s)/(1+s) √(s)√(t-s) ds)
       ≤ C(E_0+η(t)^2)e^-| + α t|^2 + y^2/M(1 + t)/(1+t)√(1+t),
    ψ(,y,t)   ≤ C e^-| + α t|^2 + y^2/M(1 + t)/1+t(E_0 + η(t)^2 ∫_0^t 1/(1+s)^2 ds) ≤ C(E_0 + η(t)^2)e^-| + α t|^2 + y^2/M(1 + t)/1+t,
    D_,y,t^𝔟ψ(,y,t)   ≤ C e^-| + α t|^2 + y^2/M(1 + t)/1+t(E_0/√(1+t) + η(t)^2 ∫_0^t 1/(1+s)^2√(1 + t - s) ds) 
       ≤ C(E_0 + η(t)^2)e^-| + α t|^2 + y^2/M(1 + t)/(1+t)√(1+t),

with ,y ∈ and 𝔟∈_≥ 0^3 with 1 ≤ |𝔟| ≤ 3. Similarly, by applying (<ref>), (<ref>) and (<ref>) to the integral system (<ref>), we bound

    _(,y,t)   ≤ C e^-| + α t|^2 + y^2/M(1 + t)/1+t(E_0 (1+1/√(t)) + η(t)^2 ∫_0^t 1/(1+s)(1+1/√(t-s)) ds)
       ≤ C(E_0+η(t)^2)e^-| + α t|^2 + y^2/M(1 + t)log(2+t)/√(1+t)√(t),
    _y(,y,t)   ≤ C e^-| + α t|^2 + y^2/M(1 + t)/1+t(E_0/√(t) + η(t)^2 ∫_0^t 1/(1+s)√(t-s) ds)
       ≤ C(E_0+η(t)^2)e^-| + α t|^2 + y^2/M(1 + t)log(2+t)/(1+t)√(t),
    _(,y,t)   ≤ C e^-| + α t|^2 + y^2/M(1 + t)/1+t(E_0 (1+1/√(t)) + ∫_0^t (1+log(2+s)/√(1+s)√(s)) (1+1/√(t-s)) ds) ≤ C,

with ,y ∈. Finally, we use ψ(s) = 0 for 0 ≤ s ≤ 1, η(t)≤ B, (<ref>), (<ref>) and (<ref>) to bound

    (,y,t)≤    v(,y,t) + (u_∞'_∞ + _(t)_∞) ψ(,y,t)≤ C(E_0 + η(t)^2)e^-| + α t|^2 + y^2/M(1 + t)/1+t,
    v_(,y,t)≤    _(,y,t) + (u_∞”_∞ + _(t)_∞)ψ(,y,t)
          +  (u_∞'_∞ + _(t)_∞) ψ_(,y,t)≤ C(E_0 + η(t)^2)e^-| + α t|^2 + y^2/M(1 + t)log(2+t)/√(t)√(1+t),

with ,y ∈. Estimate (<ref>) follows from (<ref>), (<ref>) and (<ref>), which yields the result.


 
It follows from the decomposition of the Green's function G(,,y,t) in Theorem <ref> that the leading-order linear contribution in the integral equation (<ref>) for (,y,t) comes from the term

    u_∞() ∫_^2  e(,,y-,t) v_0(,) d d.

Using the explicit formula for the Green's function e(,,y,t) given in Theorem <ref>, one readily observes from (<ref>) and (<ref>) that for v_0 ∈ W^2,∞(^2,^n) satisfying (<ref>) the optimal decay rate of (<ref>) is t^-1/2, whereas for v_0 ∈ W^1,∞(^2,^n) satisfying (<ref>) the optimal decay rate of (<ref>) is t^-1. Thus, the decay rates of (,y,t) = (,y,t) - u_∞() in Theorems <ref> and <ref> are optimal. Similarly, one can verify the optimality of the other decay rates in Theorems <ref> and <ref>. One finds that we have obtained a (possibly) suboptimal decay rate only in estimating v(,y,t) = ( + ψ(,y,t),y,t) - u_∞() in Theorem <ref>, in estimating _(,y,t) = _(,y,t) - u_∞'() in Theorem <ref> and in estimating _y(,y,t) = _y(,y,t) in Theorems <ref> and <ref>. For the first, we establish a decay rate of log(2+t)t^-1 in Theorem <ref>, whereas one would expect that the optimal decay rate is t^-1 from the estimation of the leading-order linear contribution. For the second, we establish a decay rate of log(2+t)t^-1 in Theorem <ref>, whereas one would expect t^-1. Finally, for the third,  we establish decay rates of t^-1/2 in Theorem <ref> and of log(2+t)t^-3/2 in Theorem <ref>, whereas one would expect that the optimal decay rates are t^-1 and t^-3/2, respectively. This difference of a factor t^-1/2, as well as the (possibly) artificial logarithm, might come from our approach to dealing with derivatives of the perturbation in the nonlinearity – see Remark <ref>.







§ PROOF OF POINTWISE ESTIMATES
 



 §.§ Approach
 

We prove Theorem <ref> by employing similar methods as in <cit.>, where pointwise Green's function bounds are obtained for wave trains in reaction-diffusion systems in one spatial dimension. In the proof, we distinguish between two different regimes, that is we obtain pointwise Green's function estimates for (| -  + α t|+|y|)/t large and for bounded (| -  + α t|+|y|)/t, separately.

In the first regime, we proceed as in <cit.>. We apply the Laplace transform to (<ref>) and write the solution operator e^_ν_y t as an integral

    e^_ν_y t = ∫_Γ_ν e^λ t(_ν_y - λ)^-1 dλ,

for ν_y ∈, where Γ_ν_y is a contour in the resolvent set ρ(_ν_y) containing the spectrum σ(_ν_y). Consequently, the temporal Green's function G(,,y,t) can be related via the Fourier and Laplace transforms to the resolvent kernel G_ν_y,λ(,) associated with the elliptic operator _ν_y - λ. Thus, using (<ref>) and (<ref>) we obtain

    G(,,y,t) = 1/2π i √(2π)∫_∫_Γ_ν_y e^iν_y y + λ t G_ν_y,λ(,)dλ dν_y.

We regard the eigenvalue problem _ν_yu = λ u, associated with the resolvent kernel G_ν_y,λ(,), as an ODE in the complex parameters ν_y and λ. The behavior of this ODE is well-understood when ν_y and λ are contained in a certain set ⊂^2, sufficiently far away from the origin. Thus, for (ν_y,λ) ∈ pointwise estimates on the resolvent kernel G_ν_y,λ(,) can be obtained. These estimates can then be employed to bound the temporal Green's function G(,,y,t) by deforming the contour integrals in λ and ν_y in (<ref>) to lie in . We emphasize that, in order to obtain this bound on G(,,y,t), it is crucial that (| -  + α t|+|y|)/t is sufficiently large to compensate for the fact that the resulting λ-contour lies partly in the right half-plane, far away from the origin, causing the factor e^λ t in (<ref>) to blow-up as t →∞.

In order to obtain sharp pointwise Green's function estimates for (| -  + α t|+|y|)/t bounded, it is necessary to minimize the factor e^λ t in (<ref>). This can be established by choosing the contour Γ_ν_y⊂ρ(_ν_y) to lie as much into the left half plane as possible. However, since the spectrum of _0 touches the imaginary axis at the origin, the obtained decay is not sufficient to close an eventual nonlinear iteration argument – see Remark <ref>. Instead, we proceed as in <cit.> and split off the slow-decaying translational mode first using the Fourier-Bloch decomposition (<ref>) of the solution operator. More precisely, we split (<ref>) into an integral over a small neighborhood U_0 ⊂Ω of 0, corresponding to the translational part, and a residual integral over Ω∖ U_0 to which we apply the Laplace transform. Since the resolvent set ρ(L_ν) is contained in (λ) ≤ -η for ν∈Ω∖ U_0 by (D3), we can deform the contour to obtain exponential decay from the factor e^λ t in the Laplace transform formula. Finally, we approximate the translational part by u_∞()e(,,y,t) using the expansion in Lemma <ref>.

This section is structured as follows. First, we establish pointwise estimates on the resolvent kernels associated with the operators _ν_y - λ,ν_y ∈ and L_ν - λ,ν∈Ω. Then, we recall some standard, but nontrivial, integrals that we will encounter in our estimation of the Green's function. Finally, we obtain pointwise estimates on the temporal Green's function for (| -  + α t|+|y|)/t large and (| -  + α t|+|y|)/t bounded, separately.



 §.§ Bounds on the resolvent kernel


We obtain bounds on the Green's function G_ν_y,λ(,) associated with the elliptic operator _ν_y - λ for (λ,ν_y) in a certain subset of ^2, sufficiently far away from the origin. We proceed as in <cit.> and obtain these bounds from the behavior of the spatial eigenvalues associated with the temporal eigenvalue problem _ν_y u =  λ u.

 
Complexify ν_y and consider the family _ν_y, ν_y ∈ of operators on L^2(,^n) given by (<ref>). Denote by G_ν_y,λ(,) the Green's function associated with _ν_y - λ. There exists κ_1 ∈ (0,1) and C,R,κ_2 > 1, such that for all (ν_y,λ) ∈^2 satisfying

    |ν_y| + |λ|^1/2 > R,   (λ) ≥ - κ_1(|(ν_y)|^2 + |(λ)|) + κ_2 |(ν_y)|^2,

λ is in the resolvent set ρ(L_ν_y) and the Green's function may be bounded as follows

    D_,^𝔞 G_ν_y,λ(,)   ≤ Ce^-(|λ|^1/2 + |ν_y|)|-|/C/(|λ|^1/2 + |ν_y|)^1-|𝔞|,
        ,∈,

where 𝔞∈_≥ 0^2 is a multi-index with 0 ≤ |𝔞| ≤ 1.

 Throughout this proof, we denote by C>1 a constant, which is independent of ,,λ and ν_y, that will be taken larger if necessary.

Let f ∈ L^2(,^n). We rescale the inhomogeneous problem (_ν_y - λ) u = f by setting z = (|λ|^1/2 + |ν_y|) and ϕ = (u,∂_ z u). This leads to the equivalent problem

    (∂_z - (z,ν_y,λ))ϕ = g(z,ν_y,λ),    g(z,ν_y,λ) := ([                                                  0; D^-1 f((|λ|^1/2 + |ν_y|)^-1 z)/(|λ|^1/2 + |ν_y|)^2 ]),

where

    (z,ν_y,λ) := A(ν_y,λ) + 1/|λ|^1/2 + |ν_y|B(z,ν_y,λ),    A(ν_y,λ) := ([                                       0                                      _n; k^-2(ν_y^2 + D^-1λ)/(|λ|^1/2 + |ν_y|)^2                                       0 ]),

and B is bounded on ×{(ν_y,λ) ∈^2 : |ν_y| + |λ|^1/2 > R}. Since D^-1 is symmetric and positive-definite, there exists d_min > 0 such that v^* D^-1 v ≥ d_min for all v ∈^n with v = 1. We use the latter to explicitly bound the eigenvalues of the matrix A(ν_y,λ), which are given by the square roots of the eigenvalues of its lower-left block. Thus, take κ_1 ∈ (0,1) and κ_2 > 0 such that 4κ_1D^-1 < min{2d_min,1} and 2κ_2 > 5D^-1. We find that for any (ν_y,λ) ∈^2 satisfying (<ref>) it holds

    |(σ(A(ν_y,λ)))| ≥1/2k√(min{14-D^-1κ_1,d_min2}),

i.e. A(ν_y,λ) is hyperbolic with (ν_y,λ)-independent spectral gap. We apply <cit.> (see also <cit.>) to the homogeneous problem,

    (∂_z - (z,ν_y,λ))ϕ = 0.

Thus, taking R > 1 sufficiently large, system (<ref>) has for all (ν_y,λ) ∈^2 satisfying (<ref>) an exponential dichotomy on  with constants independent of λ and ν_y. Denote by T^s,u_ν_y,λ(z,z̅) the stable and unstable evolution of (<ref>) under the exponential dichotomy. Variation of constants yields that the unique solution ϕ∈ L^2(,^n) to (<ref>) is given by

    ϕ(z,ν_y,λ) = ∫_-∞^z T^s_ν_y,λ(z,z̅) g(z̅,ν_y,λ)dz̅ - ∫_z^∞ T^u_ν_y,λ(z,z̅) g(z̅,ν_y,λ)dz̅.

Thus, rescaling back to the original coordinates, we obtain

    ([    G_ν_y,λ(,); ∂_ G_ν_y,λ(,) ])
    = ([ _n/|λ|^1/2 + |ν_y|                  0;                  0                 _n ]) · T^s_ν_y,λ((|λ|^1/2 + |ν_y|),(|λ|^1/2 + |ν_y|)) ([    0; D^-1 ]),    ≥,
    
    -T^u_ν_y,λ((|λ|^1/2 + |ν_y|),(|λ|^1/2 + |ν_y|)) ([    0; D^-1 ]),     < ,

which yields the estimates on G_ν_y,λ(,) and ∂_ G_ν_y,λ(,) in (<ref>). Finally, for the estimate on ∂_ G_ν_y,λ(,) we observe that G_ν_y,λ(,) = H_ν_y,λ(,)^*, where H_ν_y,λ(,) is the Green's function associated with the adjoint operator (_ν_y - λ)^* – see also <cit.>. Analogous to the above, one obtains

    ∂_ H_ν_y,λ(,)≤ Ce^-(|λ|^1/2 + |ν_y|)|-|/C,     ,∈,

for any (ν_y,λ) ∈^2 satisfying (<ref>) (by choosing R > 1 larger if necessary). This yields the bound on ∂_ G_ν_y,λ(,) = (∂_ H_ν_y,λ(,))^* in (<ref>). 

Using the Bloch transform, Lemma <ref> yields pointwise bounds on the resolvent kernel G_ν,λ(,) associated with the operator L_ν - λ given by (<ref>).

 
There exists constants C,R > 1 and μ > 0 such that for all (ν,λ) ∈Ω× satisfying

    |λ| > R,   (λ) ≥ -μ(ν_y^2 + |(λ)|),

λ is in the resolvent set ρ(L_ν) and the Green's function _ν,λ(,) associated with the operator L_ν - λ satisfies

    sup_ν_x ∈ [-π,π]D_,^𝔞_ν,λ(,)   ≤ C|λ|^|𝔞|-1/2,
        ,∈ [0,1],

where 𝔞∈_≥ 0^2 is a multi-index with 0 ≤ |𝔞| ≤ 1.


It holds _ν,λ(,) = e^iν_x(-)G_ν_y,λ(,) for any (ν,λ) ∈Ω× and ,∈ [0,1]. The result now follows from Lemma <ref>.

 
Let K be some compact set in Ω× such that for pairs (ν,λ) ∈ K it holds λ∈ρ(L_ν). By analyticity of _ν,λ(,) and its - and -derivatives in λ and ν there exists C>1 such that

    sup_(ν,λ) ∈ K, ,∈ [0,1]D_,^𝔞_ν,λ(,)≤ C,

where 𝔞∈_≥ 0^2 is a multi-index with 0 ≤ |𝔞| ≤ 1.




 §.§ Some useful integrals

In the pointwise estimation of the Green's function, we need to evaluate some standard, but nontrivial, integrals. First, recall

    ∫_ e^az(z-2b)dz    = √(π)/√(|a|)e^-ab^2,       a < 0, b ∈,

from <ref>. In addition, we use the identities

    ∫_ |z|^a e^-b z^2 dz    = Γ(a+1/2)b^-(a+1)/2,       a ≥ 0, b > 0, 
    ∫_0^∞e^-bz/√(z)dz    = √(π)/√(b),       b > 0.




 §.§ Pointwise Green's function estimates for large  .


As described in <ref>, we obtain pointwise bounds on the Green's function and its derivatives for large (| -  + α t|+|y|)/t by proceeding as in <cit.>.

 
Assume (D1)-(D3) hold true. There exists constants m > 0 and S,C,M>1 such that for any ,,y ∈, t > 0 satisfying

    |-+α t|+|y| ≥ St,

the Green's function G(,,y,t) associated with the operator ∂_t - enjoys the estimate

    D_,,y^𝔟 G(,,y,t)   ≤ Ct^-1-|𝔟|/2 e^-mt e^-|- + α t|^2 + |y|^2/Mt,

where 𝔟∈_≥ 0^3 is a multi-index with 0 ≤ |𝔟| ≤ 1.

 Throughout this proof, we denote by C>1 a constant, which is independent of ,,y and t, that will be taken larger if necessary.

Complexify ν_y and consider the family _ν_y, ν_y ∈ of operators analytic in ν_y given by (<ref>). Let R,κ_1,2 > 0 be as in Lemma <ref> and take

    A := ϵ_0^2 (| -  + α t| + |y|)^2/t^2,    a := sgn(y)√(A/κ_2),

where ϵ_0 > 0 is to be defined. We consider (<ref>) as a complex contour integral in ν_y. By Cauchy's integral theorem we have

    D_,^𝔞 G(,,y,t)    = 1/2π i √(2π)∫_∫_Γ_ν_y e^i(z+ia)y + λ t D_,^𝔞 G_z+ia,λ(,) dλ dz,
    ∂_y G(,,y,t)    = 1/2π√(2π)∫_∫_Γ_ν_y (z+ia) e^i(z+ia)y + λ t G_z+ia,λ(,) dλ dz,

where 𝔞∈_≥ 0^2 is a multi-index with 0 ≤ |𝔞| ≤ 1. We take the contour

    Γ_ν_y := {λ∈(λ) = A - κ_1(|(ν_y)|^2 + |(λ)|)}.

Note that for every λ∈Γ_ν_y it holds

    |ν_y|^2 + |λ| ≥max{|(λ)|, A - (λ)/κ_1}≥A/1 + κ_1.

By (<ref>) and the fact that A = κ_2 a^2, every ν_y ∈{z + ia : z ∈} and λ∈Γ_ν_y satisfy (<ref>) for A > 0 sufficiently large. Therefore, every λ∈Γ_ν_y is in the resolvent set of _ν_y for (ν_y) = a. Taking the lower bound S > 0 sufficiently large, we may assume

    |y| + |-|/t≥|y| + | -  + α t|/t - |α| ≥|y| + | -  + α t|/2t .

Finally, by taking ϵ_0 > 0 sufficiently small, we are able to estimate (<ref>) using Lemma <ref> and identities (<ref>), (<ref>), (<ref>) and (<ref>)

    D^𝔞_, G(,,y,t)   ≤ C ∫_∫_Γ_ν_y e^-a y + (λ) tD^𝔞_, G_z+ia,λ(,) dλ dz
       ≤ C ∫_∫_ e^-a y + At - √(A/1+κ_1)|-|/C - κ_1 (|(λ)| + z^2)t |(λ)|^|𝔞|-1/2 d|(λ)| dz
       ≤ C ∫_∫_ e^(ϵ_0^2 - ϵ_0/C)(|y| + | -  + α t|)^2/t - κ_1 (|(λ)| + z^2)t |(λ)|^|𝔞|-1/2 d|(λ)| dz
       ≤ Ct^-1-|𝔞|/2 e^-mt e^-|- + α t|^2 + |y|^2/Mt,

for some m > 0 and M > 1, where in the latter inequality we used that the lower bound S > 0 can be taken larger if necessary. Analogously, we estimate

    ∂_y G(,,y,t)   ≤ C ∫_∫_Γ_ν_y(|z|+|a|)e^-a y + (λ) t G_z+ia,λ(,) dλ dz
       ≤ C ∫_∫_(|z|+√(A))e^-a y + At - √(A/1+κ_1)|-|/C - κ_1 (|(λ)| + z^2)t |(λ)|^-1/2 d|(λ)| dz
       ≤ C ∫_∫_(|z|+|y| + | -  + α t|/t)e^(ϵ_0^2 - ϵ_0/C)(|y| + | -  + α t|)^2/t - κ_1 (|(λ)| + z^2)t/√(|(λ)|) d|(λ)| dz
       ≤ Ct^-3/2 e^-m t e^-|- + α t|^2 + |y|^2/Mt,

for some m > 0 and M > 1.




 §.§ Pointwise Green's function estimates for bounded  

As described in <ref>, we establish pointwise bounds on the temporal Green's function and its derivatives for bounded (| -  + α t|+|y|)/t following <cit.>.
 
Assume (D1)-(D3) hold true and let S > 1. There exists constants C,M>1 such that for any ,,y ∈, t > 0 satisfying

    |-+α t|+|y| ≤ St,

the Green's function G(,,y,t) associated with the operator ∂_t - can be decomposed as

    G(,,y,t) = 1/4π t√(d_⊥θ)e^-|-+α t|^2/4θ t - |y|^2/4d_⊥ t u_∞'() u_ ()^* + (,,y,t)

where (,,y,t) enjoys the estimates

    (,,y,t)   ≤ Ct^-1(1+t)^-1/2e^-|- + α t|^2 + |y|^2/Mt,
    D_,,y^𝔞(,,y,t)   ≤ Ct^-3/2e^-|- + α t|^2 + |y|^2/Mt,

where 𝔞∈_≥ 0^3 is a multi-index with |𝔞| = 1.

 Throughout this proof, we denote by C>1 a constant, which is independent of ,,y and t, that will be taken larger if necessary.

Without loss of generality we may assume that {ν∈^2  |ν| ≤ 2ϵ}⊂ U, where ϵ is as in (D2)-(D3) and U ⊂^2 is the neighborhood of 0 from Lemma <ref>. Let ϕ^2 → [0,1] be a smooth cut-off function satisfying ϕ(ν) = 1 if |ν| ≤ϵ and ϕ(ν) = 0 if |ν| ≥ 2ϵ. Let δ_z^m ^m → be the Dirac delta function centered at z ∈^m. Applying the Fourier-Bloch transform – see <ref> – yields 2πδ̌^2_(,0)(ν,) = e^-iν_xδ^1_(). So, using (<ref>) and Lemma <ref>, we decompose the Green's function of problem (<ref>) as

    G(,,y,t)    = [e^ tδ^2_(,0)](,y) = I + II,
    
    I := 1/4π^2∫_Ω e^iν·ξϕ(ν)P(ν)e^L_ν tδ^1_()dν,         II := 1/4π^2∫_Ω e^iν·ξ (1-ϕ(ν)P(ν))e^L_ν tδ^1_()dν,

where ξ := (-,y) and P(ν) := ⟨ q_(ν),·⟩_2 q(ν) is the spectral projection in L_^2([0,1],^n) onto (L_ν - λ_0(ν)). By Lemma <ref>, P(ν) is well-defined for any ν∈ U and depends analytically on ν. It holds P(ν) e^L_ν t = e^λ_0(ν)tP(ν) for ν∈ U. Thus, we calculate using (<ref>) and Lemma <ref>

    4π^2 I    = ∫_Ω e^iν·ξϕ(ν)P(ν)e^L_ν tδ^1_()dν
       = ∫_|ν|_^2≤ 2ϵ e^iν·ξϕ(ν)e^λ_0(ν) t q(ν,) q_(ν,)^* dν
       = ∫_^2 e^iν·ξ e^(iαν_x - θν_x^2 - d_⊥ν_y^2)t u_∞'() u_ ()^* dν - ∫_|ν|_^2≥ 2ϵ e^iν·ξ e^(iαν_x - θν_x^2 - d_⊥ν_y^2)t u_∞'() u_ ()^* dν
            + ∫_|ν|_^2≤ 2ϵ e^iν·ξ e^(iαν_x - θν_x^2 - d_⊥ν_y^2)t(e^(ν)tϕ(ν) q(ν,) q_(ν,)^* - q(0,) q_(0,)^*) dν
       = π/t√(d_⊥θ)e^-|-+α t|^2/4θ t - |y|^2/4d_⊥ t u_∞'() u_ ()^* + II' + III'.

Let 𝔟∈_≥ 0^3 with 0 ≤ |𝔟| ≤ 1. Using (<ref>) we estimate

    D^𝔟_,,y II'   ≤ C∫_|ν|_^2≥ 2ϵ(1+|ν_x| + |ν_y|)^|𝔟| e^-(θν_x^2 + d_⊥ν_y^2)t dν
       ≤ C ∫_-2ϵ^2ϵ∫_(1+|ν_x| + |ν_y|)^|𝔟| e^-θν_x^2tdν_x e^-d_⊥ν_y^2t dν_y 
            + C∫_∫_-2ϵ^2ϵ(1+|ν_x| + |ν_y|)^|𝔟| e^-θν_x^2tdν_x e^-d_⊥ν_y^2t dν_y
       ≤ Ct^-1(1 + t^-|𝔟|/2)e^-2ϵ^2 min{d_⊥,θ} t,
       ≤ C t^-1-|𝔟|/2e^-m te^-|- + α t|^2 + |y|^2/Mt,

for some m > 0 and M > 1, using that (|-+α t|+|y|)/t is bounded in the latter inequality.

The estimation of III' is more elaborate. As in <cit.> define

    a_1 :=  -  + α t/2θ t,    a_2 := y/2d_⊥ t,   ã_1,2 := sgn(a_1,2)min{2ϵ,|a_1,2|} ,

and abbreviate

    h(ν_x,ν_y,,,y,t) := e^iν·ξ e^(iαν_x - θν_x^2 - d_⊥ν_y^2)t(e^(ν)tϕ(ν) q(ν,) q_(ν,)^* - q(0,) q_(0,)^*).

We consider III' as a double complex line integral. By Cauchy's integral theorem the integral of h(·,ν_y,,,t) over [-2ϵ,2ϵ] equals the integral over the other three sides of the rectangle in  formed by the points ± 2ϵ, ± 2ϵ + iã_1. Denote by Γ_1 ⊂ the curve consisting of these three sides and define Γ_2 ⊂ similarly by the points ± 2ϵ, ± 2ϵ + iã_2. Using Cauchy's theorem twice we rewrite III' as

    D_,,y^𝔞 III'    = ∫_Γ_1∫_Γ_2 D_,,y^𝔞 h(ν_x,ν_y,,,y,t)dν_ydν_x.

Our next step is to bound the integrand D_,,y^𝔞 h(ν_x,ν_y,,,y,t) as a complex function of ν∈Γ_1 ×Γ_2. For ν∈Γ_1 ×Γ_2 ⊂ U we bound using Lemma <ref>

    e^(ν)t.   .ϕ(ν) D_,,y^𝔞(q(ν,) q_(ν,)^*) - D_,,y^𝔞(q(0,) q_(0,)^*)
       ≤|e^(ν)t - 1|ϕ(ν) D_,,y^𝔞(q(ν,) q_(ν,)^*)
            + ϕ(ν) D_,,y^𝔞(q(ν,) q_(ν,)^*) - D_,,y^𝔞(q(0,) q_(0,)^*)
       ≤ C(e^|(ν)| t - 1 + |ν|)
       ≤ C(|(ν)| t e^|(ν)|t + |ν_y| + |ν_x|)
       ≤ C((|ν_x| + |ν_y|)^3 t + |ν_y| + |ν_x|) e^c_0 ϵ (ν_x^2 + ν_y^2) t,

where c_0 is a constant independent of ϵ. By taking ϵ > 0 smaller if necessary, we may assume 2c_0ϵ≤θ, d_⊥. We apply the triangle inequality to (<ref>) and bound the integrand using (<ref>). This leads to a sum of terms of the form

    C t^m ∫_Γ_1∫_Γ_2|e^iν·ξ e^(iαν_x - θν_x^2 - d_⊥ν_y^2)t + c_0 ϵ (ν_x^2 + ν_y^2) t |ν_x|^k |ν_y|^l |dν
       = C t^m ∫_Γ_1|e^iν_x ( -  + α t) - (θ - c_0ϵ) ν_x^2 t| |ν_x|^k dν_x ∫_Γ_2|e^iν_y y - (d_⊥ - c_0ϵ) ν_y^2 t| |ν_y|^l dν_y,

with integers k, l, m ≥ 0 satisfying k + l - 2m ≥ 1. Using (<ref>) and the identities

    -ã_2y    ≤ -2d_⊥ã_2^2 t,      -zy ≤ -2d_⊥ |z| |ã_2|t ≤ -2d_⊥ z^2t,    z ∈ [0,ã_2],

we estimate the second factor in (<ref>)

    ∫_Γ_2|e^iν_y y - (d_⊥ - c_0ϵ) ν_y^2 t| |ν_y|^l dν_y,
         ≤ C ∫_-2ϵ^2ϵ|e^i(z + iã_2) y - (d_⊥ - c_0ϵ) (z+iã_2)^2t| |z+iα̃_̃2̃|^l dz + C∫_0^ã_2|e^i(2ϵ + iz) y - (d_⊥ - c_0ϵ) (2ϵ + iz)^2t| |2ϵ + iz|^l dz 
         ≤ C e^-d_⊥ã_2^2 t/2∑_j = 0^l∫_-2ϵ^2ϵ e^-(d_⊥ - c_0ϵ)z^2 t |z|^l-j|α̃_̃2̃|^j dz + Ce^-2d_⊥ϵ^2 t∑_j = 0^l∫_0^|ã_2| e^-(d_⊥ - c_0ϵ) z^2 t |z|^l-jϵ^j dz 
         ≤ C  e^-d_⊥ã_2^2 t/2∑_j = 0^lmin{1,t^-(l+1-j)/2 |α̃_̃2̃|^j} + e^-2d_⊥ϵ^2 t/2∑_j = 0^l min{1, t^-(l+1-j)/2ϵ^j}
         ≤ C  e^-d_⊥ã_2^2 t/4min{1, t^-(l+1)/2} + e^-d_⊥ϵ^2 tmin{1, t^-(l+1)/2}
         ≤ C (1+t)^-(1+l)/2(e^-d_⊥ã_2^2 t/4 + e^-d_⊥ϵ^2 t)
         ≤ C (1+t)^-(1+l)/2 e^-|y|^2/Mt,

for some M > 0, using that (|-+α t|+|y|)/t is bounded in the latter inequality. Similarly, one establishes the bound on the second factor in (<ref>)

    ∫_Γ_1|e^iν_x ( -  + α t) - (θ - c_0ϵ) ν_x^2 t| |ν_x|^k dν_x ≤ C (1+t)^-(1+k)/2 e^-|- + α t|^2/Mt,

for some M > 0. We employ the latter two estimates to bound the summands of the form (<ref>) in order to obtain

    D_,,y^𝔞 III'   ≤∫_Γ_1∫_Γ_2 D_,,y^𝔞 h(ν_x,ν_y,,,y,t)dν_ydν_x ≤ C(1+t)^-3/2 e^-|-+α t|^2 + |y|^2/Mt,

using k+l-2m≥ 1 holds in (<ref>).

Next, we split II into two integrals

    4π^2 II    = ∫_|ν|_^2≤ϵ e^iν·ξ (1-ϕ(ν)P(ν))e^L_ν tδ̌_(ν,)dν + ∫_ν∈Ω, |ν|_^2≥ϵ e^iν·ξ (1-ϕ(ν)P(ν))e^L_ν tδ̌_(ν,)dν
       = I + II,

We start estimating II. Note that by assumption (D3) the spectrum σ(L_ν) is confined to (λ) < -η for ν∈Ω with |ν| ≥ϵ. Combining this with Corollary <ref> yields that there exists κ > 0, independent of ν and λ, such that the contour

    Γ_ν := {λ∈(λ) = -κ(1 + ν_y^2 + |(λ)|)},

is contained in the resolvent set of L_ν for ν∈Ω with |ν| ≥ϵ. Using the Laplace transform we rewrite II as

    II = 1/2π i∫_ν∈Ω, |ν| ≥ϵ∫_Γ_ν e^iν·ξ + λ t (1-ϕ(ν)P(ν)) _ν,λ(,)dλ dν,

where _ν,λ(,) denotes the Green's function associated with the operator L_ν - λ. Let j ∈{0,1}. Using Corollary <ref>, Remark <ref> and the integral identities (<ref>) and (<ref>) we estimate

    ∂_y^j  II    ≤ C ∫_ν∈Ω, |ν| ≥ϵ∫_Γ_ν |ν_y|^j e^(λ) t_ν,λ(,)dλ dν
       ≤ C e^-κ t∫_∫_0^∞|ν_y|^j/√(|(λ)|) e^-κ t(|(λ)|  + ν_y^2)d|(λ)| dν_y
       ≤ C t^-1- j/2 e^-κ t
       ≤ C t^-1 - j/2e^-m te^-|- + α t|^2 + |y|^2/Mt,

for some m > 0 and M > 1, using that (|-+α t|+|y|)/t is bounded in the latter inequality. Let 𝔠∈_≥ 0^2 be a multi-index with |𝔠| = 1. Analogously, we estimate

    D_,^𝔠 II    ≤ C ∫_ν∈Ω, |ν| ≥ϵ∫_Γ_ν e^(λ) t(|ν_x|_ν,λ(,)+D_,^𝔠_ν,λ(,))dλ dν
       ≤ C e^-κ t∫_∫_0^∞(1 + 1/√(|(λ)|)) e^-κ t(|(λ)|  + ν_y^2)d|(λ)| dν_y
       ≤ C t^-1(1 + t^-1/2) e^-κ t
       ≤ C t^-3/2 e^-m te^-|- + α t|^2 + |y|^2/Mt,

for some m > 0 and M > 1.

For the estimation of I we note that ϕ(ν) = 1 for |ν|_^2≤ϵ. For |ν|_^2≤ϵ the spectrum of the operator L_ν restricted to (P(ν)) is confined to (λ) < -η by Lemma <ref> (by taking η smaller if necessary), since P(ν) is the spectral projection associated with the critical eigenvalue λ_0(ν) of L_ν. Therefore, we may assume (by adapting κ > 0 if necessary) that the contour Γ_ν defined in (<ref>) lies in the resolvent set of L_ν|_(P(ν)) for all |ν|_^2≤ϵ by Corollary <ref>. Estimating I is therefore similar to estimating II. Let _ν,λ(,) be the Green's function associated with the operator L_ν|_(P(ν)) - λ. Let K be some compact set in Ω× such that for pairs (ν,λ) ∈ K it holds λ∈ρ(L_ν|_(P(ν))). By analyticity of the Green's function in λ and ν it holds

    sup_[ (ν,λ) ∈ K,;   ,∈ [0,1] ]D_,^𝔠_ν,λ(,)≤ C,

where 𝔠∈_≥ 0^2 is a multi-index with 0 ≤ |𝔠| ≤ 1. Thus, using the Laplace transform, Corollary <ref> and the integral identities (<ref>) and (<ref>) we estimate

    ∂_y^j  I    ≤ C ∫_|ν|_^2≤ϵ∫_λ∈Γ_ν, |λ| ≤ R |ν_y|^j e^(λ) t_ν,λ(,)dλ dν
            + C∫_|ν|_^2≤ϵ∫_λ∈Γ_ν, |λ| > R |ν_y|^j e^(λ) t_ν,λ(,)dλ dν
       ≤ Ce^-κ t∫_∫_0^∞ |ν_y|^j  (1 + 1/√(|(λ)|))  e^-κ t(|(λ)|  + ν_y^2)d|(λ)| dν_y
       ≤ C t^-1 - j/2e^-m te^-|- + α t|^2 + |y|^2/Mt,

for some m > 0 and M > 1, using that (|-+α t|+|y|)/t is bounded in the latter inequality. Analogously, we estimate

    D_,^𝔠 I    ≤ C ∫_|ν|_^2≤ϵ∫_λ∈Γ_ν, |λ| ≤ R e^(λ) t(|ν_x|_ν,λ(,) + D_,^𝔠_ν,λ(,))dλ dν
            + C∫_|ν|_^2≤ϵ∫_λ∈Γ_ν, |λ| > R e^(λ) t(|ν_x|_ν,λ(,)+D_,^𝔠_ν,λ(,))dλ dν
       ≤ C e^-κ t∫_∫_0^∞(1 + 1/√(|(λ)|)) e^-κ t(|(λ)|  + ν_y^2)d|(λ)| dν_y
       ≤ C t^-3/2e^-m te^-|- + α t|^2 + |y|^2/Mt,

for some m > 0 and M > 1.




 §.§ Proof of Theorem <ref>


We employ Lemmas <ref> and <ref> to prove Theorem <ref>. First, observe that a direct calculation establishes the estimates in (<ref>) on e(,,y,t) and its derivatives. Now, let S > 1 be as in Lemma <ref>. It holds by Lemma <ref>

    (,,y,t) = (,,y,t) + 1-χ(t)/4π t√(d_⊥θ)e^-|-+α t|^2/4θ t - |y|^2/4d_⊥ t u_∞'() u_ ()^*    if  |-+α t|+|y| ≤ St,
    
    G(,,y,t) - e(,,y,t)    if  |-+α t|+|y| > St,

for ,,y ∈ and t > 0. The bounds in (<ref>) on G(,,y,t) and its derivatives follow for ,,y ∈, t > 0 satisfying |-+α t|+|y| ≤ St  from Lemma <ref> and the fact that t ↦ 1 - χ(t) vanishes for t ≥ 2. The bounds in (<ref>) on G(,,y,t) and its derivatives follow for ,,y ∈, t > 0 satisfying |-+α t|+|y| > St from Lemma <ref> and the estimate

    D_,,y^𝔞 e(,,y,t)   ≤ C(1+t)^-1 e^-|- + α t|^2 + |y|^2/Mt≤ C(1+t)^-1e^-S^2/2Mt e^-|- + α t|^2 + |y|^2/2Mt,

where C,M > 1 are some constants, which are independent of ,,y and t. 

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