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This paper is concerned with the Cauchy problem $$u_t=u_{xx} +f(t,u), \,\,\,x\in\mathbb{R},\,t>0, $$ $$u(0,x)= u_0(x), \,\,\, x\in\mathbb{R},$$ where $f$is a rather general nonlinearity that is periodic in $t$, and satisfies$f(\cdot,0)\equiv 0$ and that the corresponding ODE has a positive periodicsolution $p(t)$. Assuming that $u_0$ is front-like, that is, $u_0(x)$ is closeto $p(0)$ for $x\approx -\infty$ and close to $0$ for $x\approx \infty$, we aimto determine the long-time dynamical behavior of the solution $u(t,x)$ by usingthe notion of propagation terrace introduced by Ducrot, Giletti and Matano(2014). We establish the existence and uniqueness of propagating terrace for avery large class of nonlinearities, and show the convergence of the solution$u(t,x)$ to the terrace as $t\to\infty$ under various conditions on $f$ or$u_0$. We first consider the special case where $u_0$ is a Heaviside typefunction, and prove the converge result without requiring any non-degeneracy on$f$. Furthermore, if $u_0$ is more general such that it can be trapped betweentwo Heaviside type functions, but not necessarily monotone, we show that theconvergence result remains valid under a rather mild non-degeneracy assumptionon $f$. Lastly, in the case where $f$ is a non-degenerate multistablenonlinearity, we show the global and exponential convergence for a much largerclass of front-like initial data.

Dynamics of time-periodic reaction-diffusion equations with front-like initial data on $\mathbb{R}$