Long time behavior of random and nonautonomous Fisher-KPP equations. Part II. Transition fronts

Abstract

In the current series of two papers, we study the long time behavior of the following random Fisher-KPP equation $$ u_t =u_{xx}+a(\theta_t\omega)u(1-u),\quad x\in\R, \eqno(1) $$ where $\omega\in\Omega$, $(\Omega, \mathcal{F},\mathbb{P})$ is a given probability space, $\theta_t$ is an ergodic metric dynamical system on $\Omega$, and $a(\omega)>0$ for every $\omega\in\Omega$. We also study the long time behavior of the following nonautonomous Fisher-KPP equation, $$ u_t=u_{xx}+a_0(t)u(1-u),\quad x\in\R, \eqno(2) $$ where $a_0(t)$ is a positive locally H\"older continuous function. In the first part of the series, we studied the stability of positive equilibria and the spreading speeds of (1) and (2). In this second part of the series, we investigate the existence and stability of transition fronts of (1) and (2). We first study the transition fronts of (1). Under some proper assumption on $a(\omega)$, we show the existence of random transition fronts of (1) with least mean speed greater than or equal to some constant $\underline{c}^*$ and the nonexistence of ranndom transition fronts of (1) with least mean speed less than $\underline{c}^*$. We prove the stability of random transition fronts of (1) with least mean speed greater than $\underline{c}^*$. Note that it is proved in the first part that $\underline{c}^*$ is the infimum of the spreading speeds of (1). We next study the existence and stability of transition fronts of (2). It is not assumed that $a(\omega)$ and $a_0(t)$ are bounded above and below by some positive constants. Many existing results in literature on transition fronts of Fisher-KPP equations have been extended to the general cases considered in the current paper. The current paper also obtains several new results.