PDE Seminar Abstracts

Spreading and Vanishing in Nonlinear Diffusion Problems with Free Boundaries

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


We consider nonlinear diffusion problems of the form ut = uxx + 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 C1 and satisfies f(0) = 0, we show that every bounded positive solution converges to a stationary solution as t . 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 σ in the initial data, we reveal a threshold value σ* such that spreading ( lim tu = 1) happens when σ > σ*, vanishing ( lim tu = 0) happens when σ < σ*, and at the threshold value σ*, lim tu 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.