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

# 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 Cîrstea or Daniel Daners.

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