**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
Cîrstea or Daniel Daners.