Spreading and vanishing in nonlinear diffusion problems with free boundaries

Abstract

We study nonlinear diffusion problems of the form $u_t=u_{xx}+f(u)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special $f(u)$ of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any $f(u)$ which is $C^1$ and satisfies $f(0)=0$, we show that the omega limit set $\omega (u)$ of every bounded positive solution is determined by a stationary solution. For monostable, bistable and combustion types of nonlinearities, we obtain a rather complete description of the long-time dynamical behavior of the problem; moreover, by introducing a parameter $\sigma$ in the initial data, we reveal a threshold value ${\sigma }^{\ast }$ such that spreading (${\lim }_{t\to \infty }u=1$) happens when $\sigma >{\sigma }^{\ast }$, vanishing (${\lim }_{t\to \infty }u=0$) happens when $\sigma <{\sigma }^{\ast }$, and at the threshold value ${\sigma }^{\ast }$, $\omega (u)$ is different for the three different types of nonlinearities. When spreading happens, we make use of "semi-waves" to determine the asymptotic spreading speed of the front.