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

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