Stability of transition waves and positive entire solutions of Fisher-KPP equations with time and space dependence

TL;DR

This paper proves the existence, uniqueness, and stability of positive solutions and transition waves in Fisher-KPP equations with general time and space dependence, including periodic and almost periodic cases, establishing their asymptotic stability.

Contribution

First to analyze the stability of transition waves in Fisher-KPP equations with both time and space dependence, extending previous work to more general settings.

Findings

Existence and uniqueness of positive entire solutions

Stability of transition waves connecting solutions and zero

Transition waves are asymptotically stable under broad conditions

Abstract

This paper is concerned with the stability of transition waves and strictly positive entire solutions of random and nonlocal dispersal evolution equations of Fisher-KPP type with general time and space dependence, including time and space periodic or almost periodic dependence as special cases. We first show the existence, uniqueness, and stability of strictly positive entire solutions of such equations. Next, we show the stability of uniformly continuous transition waves connecting the unique strictly positive entire solution and the trivial solution zero and satisfying certain decay property at the end close to the trivial solution zero (if it exists). The existence of transition waves has been studied in [34, 39, 45, 46, 61] for random dispersal Fisher-KPP equations with time and space periodic dependence, in [41, 42, 43, 51, 52, 53, 58, 63] for random dispersal Fisher-KPP equations with quite general time and/or space dependence, and in [17, 48, 56] for nonlocal dispersal Fisher-KPP equations with time and/or space periodic dependence. The stability result established in this paper implies that the transition waves obtained in many of the above mentioned papers are asymptotically stable. Up to the author's knowledge, it is the first time that the stability of transition waves of Fisher-KPP equations with both time and space dependence is studied.