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.