Propagating Terrace and Asymptotic Profile to Time-Periodic Reaction-Diffusion Equations

This paper is concerned with the asymptotic behavior of solutions of time periodic reaction-diffusion equation \begin{equation*}\label{aaa} \begin{cases} u_{t}(x,t)=u_{xx}(x,t)+f(t,u(x,t)),\quad \,\,\forall x\in\mathbb{R},\,t>0,\\ u(x,0)=u_{0}(x), \quad \quad\quad\quad\quad\quad\quad\quad\quad \forall x\in\mathbb{R}, \end{cases} \end{equation*} where $u_{0}(x)$ is the Heaviside type initial function and $f(t,u)$ satisfies $f(T+t,u)=f(t,u)$. Under certain conditions, we prove that there exists a minimal propagating terrace (a family of pulsating traveling fronts) in some specific sense and the solution of the above equation converges to the minimal propagating terrace.