Spreading and Vanishing in Nonlinear Diffusion Problems with Free Boundaries

Yihong Du
The University of New England
19 Sep 2011, 2-3pm, Eastern Avenue Seminar Room 405

Abstract

We consider 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 any $f(u)$ which is $C^1$ and satisfies $f(0)=0$, we show that every bounded positive solution converges to a stationary solution as $t\to \mathrm{\infty }$. 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 ${\sigma }^{\ast }$ such that spreading ($\underset{t\to \mathrm{\infty }}{lim}u=1$) happens when $\sigma >{\sigma }^{\ast }$, vanishing ($\underset{t\to \mathrm{\infty }}{lim}u=0$) happens when $\sigma <{\sigma }^{\ast }$, and at the threshold value ${\sigma }^{\ast }$, $\underset{t\to \mathrm{\infty }}{lim}u$ 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.

Check also the PDE Seminar page. Enquiries to Florica C�rstea or Daniel Daners.