Propagation phenomena for time heterogeneous KPP reaction-diffusion equations

TL;DR

This paper studies propagation phenomena in time-heterogeneous KPP reaction-diffusion equations, establishing the existence of generalized and random transition waves, and analyzing spreading properties of solutions.

Contribution

It introduces the existence of generalized transition waves for time-dependent KPP equations and extends results to random stationary ergodic cases.

Findings

Existence of generalized transition waves for certain speeds.

Existence of random transition waves under ergodic conditions.

Spreading properties of solutions to the Cauchy problem.

Abstract

We investigate in this paper propagation phenomena for the heterogeneous reaction-diffusion equation $\partial_t u -\Delta u = f(t,u)$, $x\in R^N$, $t\in\R$, where f=f(t,u) is a KPP monostable nonlinearity which depends in a general way on t. A typical f which satisfies our hypotheses is f(t,u)=m(t) u(1-u), with m bounded and having positive infimum. We first prove the existence of generalized transition waves (recently defined by Berestycki and Hamel, Shen) for a given class of speeds. As an application of this result, we obtain the existence of random transition waves when f is a random stationary ergodic function with respect to t. Lastly, we prove some spreading properties for the solution of the Cauchy problem.