Spreading and Vanishing in Nonlinear Diffusion Problems with Free Boundaries

Abstract

We consider nonlinear diffusion problems of the form $u_t=u_{xx}+f\left(u\right)$ 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 any $f\left(u\right)$ which is $C^1$ and satisfies $f\left(0\right)=0$, we show that every bounded positive solution converges to a stationary solution as $t\to \infty$. For monostable, bistable and combustion types of nonlinearities, we obtain a 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 ($\underset{t\to \infty }{lim}u=1$) happens when $\sigma >{\sigma }^{*}$, vanishing ($\underset{t\to \infty }{lim}u=0$) happens when $\sigma <{\sigma }^{*}$, and at the threshold value ${\sigma }^{*}$, $\underset{t\to \infty }{lim}u$ 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.