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

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.

arXiv comment32pages

References in corpus (2)

• Radial terrace solutions and propagation profile of multistable reaction-diffusion equations over ${\mathbb{R}}^N$
• Dynamics of Time-Periodic Reaction-Diffusion Equations with Compact Initial Support on R

Cited by in corpus (1)

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