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

TL;DR

This paper investigates the long-term behavior of solutions to time-periodic reaction-diffusion equations, establishing the existence of a minimal propagating terrace and showing solutions converge to this structure over time.

Contribution

The paper introduces the concept of a minimal propagating terrace for time-periodic reaction-diffusion equations and proves convergence of solutions to this terrace.

Findings

Existence of a minimal propagating terrace under certain conditions

Convergence of solutions to the minimal propagating terrace

Characterization of pulsating traveling fronts

Abstract

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.