Traveling fronts guided by the environment for reactiondiffusion equations
Henri Berestycki  CAMS, UMR 8557, EHESS, 190198 avenue de France, 75244 Paris Cedex 13, France (email) Abstract:
This paper deals with the existence of traveling fronts for the
reactiondiffusion equation:
$$
\frac{\partial u}{\partial t}  \Delta u =h(u,y) \qquad t\in \mathbb{R}, \; x=(x_1,y)\in \mathbb{R}^N.
$$
We first consider the case $h(u,y)=f(u)\alpha g(y)u$ where $f$ is of KPP or bistable type and $\lim_{y\rightarrow +\infty}g(y)=+\infty$. This equation comes from a model in population dynamics in which there is spatial spreading as well as phenotypic mutation of a quantitative phenotypic trait that has a locally preferred value. The goal is to understand spreading and invasions in this heterogeneous context. We prove the existence of threshold value $\alpha_0$ and of a nonzero asymptotic profile (a stationary limiting solution) $V(y)$ if and only if $\alpha<\alpha_0$. When this condition is met, we prove the existence of a traveling front. This allows us to completely identify the behavior of the solution of the parabolic problem in the KPP case.
Keywords: Traveling front, reactiondiffusion, KPP, population dynamics.
Received: May 2012; Revised: March 2013; Available Online: April 2013. 
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