Traveling fronts in space-time periodic media

TL;DR

This paper investigates the existence and properties of pulsating traveling fronts in space-time periodic media, establishing conditions for their speeds, and characterizing these speeds in specific reaction scenarios like KPP-type and concave reactions.

Contribution

It proves the existence of critical speeds for pulsating fronts in periodic media and characterizes these speeds using eigenvalues, with special results for KPP and concave reaction terms.

Findings

Existence of speeds c* and c** for pulsating fronts

Equality c* = c** in KPP-type reactions and eigenvalue characterization

Lipschitz continuity of front profiles when reaction is concave

Abstract

This paper is concerned with the existence of pulsating traveling fronts for the equation: $\partial_t u - \nabla \cdot (A(t, x)\nabla u) + q(t, x) \cdot \nabla u = f (t, x, u)$, (1) where the diffusion matrix $A$, the advection term $q$ and the reaction term $f$ are periodic in $t$ and $x$. We prove that there exist some speeds $c^*$ and $c^{**}$ such that there exists a pulsating traveling front of speed $c$ for all $c\ge c^{**}$ and that there exists no such front of speed $c<c^*$. We also give some spreading properties for front-like initial data. In the case of a KPP-type reaction term, we prove that $c^* = c^{**}$ and we characterize this speed with the help of a family of eigenvalues associated with the equation. If $f$ is concave with respect to $u$, we prove some Lipschitz continuity for the profile of the pulsating traveling front. Cet articl{\'e} etudie l'existence de fronts pulsatoires pour l'\'equation : $\partial_t u - \nabla \cdot (A(t, x)\nabla u) + q(t, x) \cdot \nabla u = f (t, x, u)$, (2) o\`u la matrice de diffusion A, le terme d'advection $q$ et le terme de r{\'e}action $f$ sont p{\'e}riodiques en $t$ et en $x$. Nous prouvons l'existence de deux vitesses $c^*$ et $c^{**}$ telles qu'il existe un front pulsatoire de vitesse $c$ pour tout $c \ge c^{**}$ et qu'il n'existe pas de tel front de vitesse $c<c^*$. Nous donnons egalement des propri{\'e}t{\'e}s de spreading pour des donn{\'e}es initiales ressemblant a des fronts. Dans le cas d'un terme de r{\'e}action de type KPP, nous prouvons que $c^* = c^{**}$ et nous caract{\'e}risons cette vitess{\`e} a l'aide d'une famille de valeurs propres associ{\'e}{\`e} a l'\'equation. Si $f$ est concave en $u$, nous montrons que le profil du front pulsatoire construit est lipschitzien.