Spreading properties and forced traveling waves of reaction-diffusion equations in a time-heterogeneous shifting environment

In this paper, we study the propagation dynamics for a large class of time and space heterogeneous reaction-diffusion equations $u_t=u_{xx}+ug(x-\omega (t),t,u)$, where $\omega \left(t\right)$ represents the shifting distance, and the nonlinearity $ug(\xi ,t,u)$ is asymptotically of KPP type as $\xi \to -\infty$ and is negative as $\xi \to +\infty$. Let $c^{⁎}$ be the spreading speed of the limiting equation $u_t=u_{xx}+ug(-\infty ,t,u)$. Under the assumption that the shifting speed ${\omega }^{\prime }\left(t\right)$ admits a uniform mean c, we show that the solutions with compactly supported initial data go to zero eventually when $c\le -c^{⁎}$, the leftward spreading speed is $-c^{⁎}$ when $c>-c^{⁎}$, and the rightward spreading speed is c and $c^{⁎}$ when $c\in (-c^{⁎},c^{⁎})$ and $c\ge c^{⁎}$, respectively. We also establish the existence, uniqueness and nonexistence of the forced traveling wave in terms of the sign of $c-c^{⁎}$.