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

Abstract

In this paper, we define the transition semi-wave solution (cf. Definition 1.1) of the following reaction diffusion equation with free boundaries(1){ut=uxx+f(t,x,u),t∈R,x<h(t),u(t,h(t))=0,t∈R,h′(t)=−μux(t,h(t)),t∈R. In the homogeneous case, i.e., f(t,x,u)=f(u), under the hypothesisf(u)∈C1([0,1]),f(0)=f(1)=0,f′(1)<0,f(u)<0foru>1, we prove that the semi-wave connecting 1 and 0 of (1) 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 ut=u(c(t)−u) (resp. uxx+u(a(x)−u)=0) and 0 is exactly the unique almost periodic semi-wave of (1). Finally, we provide an example of the heterogeneous equation to show the existence of the transition semi-wave without any global mean speeds.