Maximal and minimal spreading speeds for reaction diffusion equations in nonperiodic slowly varying media

TL;DR

This paper studies the long-term spreading speeds of solutions to a heterogeneous Fisher-KPP equation, revealing how the growth rate of the medium's heterogeneity influences whether the speed oscillates or converges to a unique value.

Contribution

It characterizes the asymptotic spreading speeds in nonperiodic media with slowly varying heterogeneity, distinguishing between oscillatory and unique speeds based on the growth rate of the heterogeneity.

Findings

Spreading speed oscillates between two values when heterogeneity grows slowly.

A unique propagation speed is explicitly computed for rapidly growing heterogeneity.

The behavior depends on the rate at which the heterogeneity function approaches infinity.

Abstract

This paper investigates the asymptotic behavior of the solutions of the Fisher-KPP equation in a heterogeneous medium, $$\partial_t u = \partial_{xx} u + f(x,u),$$ associated with a compactly supported initial datum. A typical nonlinearity we consider is $f(x,u) = \mu_0 (\phi (x)) u(1-u)$, where $\mu_0$ is a 1-periodic function and $\phi$ is a $\mathcal{C}^1$ increasing function that satisfies $\lim_{x\to +\infty} \phi (x) = +\infty$ and $\lim_{x\to +\infty} \phi' (x) = 0$. Although quite specific, the choice of such a reaction term is motivated by its highly heterogeneous nature. We exhibit two different behaviors for $u$ for large times, depending on the speed of the convergence of $\phi$ at infinity. If $\phi$ grows sufficiently slowly, then we prove that the spreading speed of $u$ oscillates between two distinct values. If $\phi$ grows rapidly, then we compute explicitly a unique and well determined speed of propagation $w_\infty$, arising from the limiting problem of an infinite period. We give a heuristic interpretation for these two behaviors.