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 \(t\to\infty\). 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 (\(\lim_{t\to\infty}u= 1\)) happens when \(\sigma>\sigma^*\), vanishing (\(\lim_{t\to\infty}u=0\)) happens when \(\sigma<\sigma^*\), and at the threshold value \(\sigma^*\), \(\lim_{t\to\infty}u\) 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 Cîrstea or Daniel Daners.