KIT

Abstract

We will present in this talk propagation properties for the solutions of the heterogeneous Fisher-KPP equation $$\partial_{t} u - \partial_{xx}u=\mu (t,x) u(1-u)$$ where $\mu$ is only assumed to be uniformly continuous and bounded in $(t,x)$, for initial data with compact support. Using homogenization techniques, we construct two speeds $\overline{w}$ and $\underline{w}$ such that $\lim_{t\to+\infty}u(t,x+wt) = 0$ if $w>\overline{w}$ and $\lim_{t\to+\infty} u(t,x+wt)=1$ if $w<\underline{w}$. These speeds are characterized in terms of two new notions of generalized principal eigenvalues for linear parabolic operators in unbounded domains. In particular, this allows us to derive the exact asymptotic speed of propagation for almost periodic and asymptotically almost periodic equations (where $\overline{w}=\underline{w}$) and to obtain explicit bounds on these speeds in recurrent and spatially homogeneous equations.