Asymptotic Behavior of Solutions of Free Boundary Problems for Fisher-KPP Equation

Abstract

We study a free boundary problem for Fisher-KPP equation: $$u_t=u_{xx}+f(u)$$ ( $$g(t)< x < h(t)$$ ) with free boundary conditions $$h'(t)=-u_x(t,h(t))-\beta $$ and $$g'(t)=-u_x(t,g(t))-\alpha $$ for $$\alpha >0$$ and $$\beta \in \mathbb {R}$$ . Such a free boundary problem can model the spreading of a biological or chemical species affected by the boundary environment. $$\beta >0$$ means that there is a “resistance force” with strength $$\beta $$ at boundary $$x=h(t)$$ . $$\beta <0$$ (resp. $$\alpha >0$$ ) means that there is an enhancing force with strength $$\beta $$ (resp. $$\alpha $$ ) at the boundary $$x=h(t)$$ (resp. g(t)). There are many parts of $$(\alpha ,\beta )$$ . In different parts, the asymptotic behavior of solutions are different. In the first part, we have a spreading-transition-vanishing result: either spreading happens (the solution converges to 1 in the moving frame), or in the transition case (the solution will converge to the compactly supported traveling wave), or vanishing happens (the solution converges to 0 within a finite time). In the second part, we also have a trichotomy result, but in transition case the solution will converge to the non-monotonous traveling semi-wave, and the vanishing case has three different types. For the third part, only spreading happens for any solution. In the fourth part ( $$\alpha $$ or $$\beta $$ large), any solution will vanish, also there are three types of vanishing. For the case $$\alpha = \beta $$ , we have two different trichotomy results and a dichotomy result.