Propagation dynamics of a reaction-diffusion equation in a time-periodic shifting environment

Abstract

This paper concerns the nonautonomous reaction-diffusion equation \[ u_t=u_{xx}+ug(t,x-ct,u), \quad t>0,x\in\mathbb{R}, \] where $c\in \mathbb{R}$ is the shifting speed, and the time periodic nonlinearity $ug(t,\hat{I}¾,u)$ is asymptotically of KPP type as $\hat{I}¾\to -\infty$ and is negative as $\hat{I}¾\to +\infty$. Under a subhomogeneity condition, we show that there is $c^{\ast }>0$ such that a unique forced time periodic wave exists if and only $|c|<c^{\ast }$ and it attracts other solutions in a certain sense according to the tail behavior of initial values. In the case where $|c|\ge c^{\ast }$, the propagation dynamics resembles that of the limiting system as $\hat{I}¾\to \pm \infty$, depending on the shifting direction.

28 pages