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

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.