Transition semi-wave solutions of reaction diffusion equations with free boundaries

TL;DR

This paper defines and analyzes transition semi-wave solutions for reaction diffusion equations with free boundaries, proving uniqueness and characterizing all bounded solutions in various homogeneous and almost periodic cases.

Contribution

It introduces the concept of transition semi-waves for free boundary problems and establishes their uniqueness and characterization in homogeneous and almost periodic settings.

Findings

Unique semi-wave connecting 1 and 0 when it exists.

All bounded transition semi-waves are exactly the semi-wave.

Existence of transition semi-waves without global mean speeds.

Abstract

In this paper, we define the transition semi-wave solution of the following reaction diffusion equation with free boundaries \begin{equation}\label{0.1} \left\{ \begin{aligned} u_{t}=u_{xx}+f(t,x,u),\ \ &t\in\Real, x<h(t), u(t,h(t))=0,\ \ &t\in\Real, h^{\prime}(t)=-\mu u_{x}(t,h(t)),\ \ &t\in\Real, \end{aligned} \right. \end{equation} In the homogeneous case, i.e., $f(t,x,u)=f(u)$, under the hypothesis $$ f(u)\in {C}^{1}([0,1]), f(0)=f(1)=0, f^{\prime}(1)<0, f(u)<0\ \text{for}\ u>1, $$ we prove that the semi-wave connecting $1$ and $0$ is unique provided it exists. Furthermore, we prove that any bounded transition semi-wave connecting $1$ and 0 is exactly the semi-wave. In the cases where $f$ is KPP-Fisher type and almost periodic in time (space), i.e., $f(t,x,u)=u(c(t)-u)$ (resp. $u(a(x)-u)$) with $c(t)$ (resp. $a(x)$) being almost periodic, applying totally different method, we also prove any bounded transition semi-wave connecting the unique almost periodic positive solution of $u_{t}=u(c(t)-u)$ (resp. $u_{xx}+u(a(x)-u)=0$) and $0$ is exactly the unique almost periodic semi-wave. Finally, we provide an example of the heterogeneous equation to show the existence of the transition semi-wave without any global mean speeds.