Traveling fronts guided by the environment for reaction-diffusion equations

This paper deals with the existence of traveling fronts for the reaction-diffusion equation: ∂u ∂t −∆u = h(u, y) t ∈ R, x = (x1, y) ∈ R . We first consider the case h(u, y) = f(u) − αg(y)u where f is of KPP or bistable type and lim|y|→+∞ g(y) = +∞. This equation comes from a model in population dynamics in which there is spatial spreading as well as genetic mutation of a quantitative genetic 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 α0 and of a nonzero asymptotic profile (a stationary limiting solution) V (y) if and only if α < α0. 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. We also study here the case where h(y, u) = f(u) for |y| ≤ L1 and h(y, u) ≈ −αu for |y| > L2 ≥ L1. This equation provides a general framework for a model of cortical spreading depressions in the brain. We prove the existence of traveling front if L1 is large enough and the non-existence if L2 is too small.