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

This paper concerns the nonautonomous reaction-diffusion equation $u_t=u_{xx}+ug(t,x-ct,u), \quad t>0,x\in\mathbb{R},$ where $c\in\mathbb{R}$ is the shifting speed, and the time periodic nonlinearity $ug(t,ξ,u)$ is asymptotically of KPP type as $ξ\to-\infty$ and is negative as $ξ\to+\infty$. Under a subhomogeneity condition, we show that there is $c^*>0$ 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 $|c|\ge c^*$, the propagation dynamics resembles that of the limiting system as $ξ\to\pm \infty$, depending on the shifting direction.

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