SMS scnews item created by Daniel Daners at Thu 15 Sep 2011 1313
Type: Seminar
Modified: Fri 16 Sep 2011 1153; Fri 16 Sep 2011 1208
Distribution: World
Expiry: 19 Sep 2011
Calendar1: 19 Sep 2011 1400-1500
CalLoc1: Eastern Ave. Room 405
Auth: daners@bari.maths.usyd.edu.au

PDE Seminar

Spreading and Vanishing in Nonlinear Diffusion Problems with Free Boundaries

Du

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 CÓrstea or Daniel Daners.


If you are registered you may mark the scnews item as read.
School members may try to .