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 [8]. 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 $\sigma^*$ such that spreading ($\lim_{t\to\infty}u= 1$) happens when $\sigma>\sigma^*$, vanishing ($\lim_{t\to\infty}u=0$) happens when $\sigma<\sigma^*$, and at the threshold value $\sigma^*$, $\omega(u)$ 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.