Convergence of bounded solutions of nonlinear parabolic problems on a bounded interval: the singular case

Daniel Hauer
University of Sydney
8 April 2013 14:00-15:00, Carslaw Room 829 (AGR)

Abstract

In this talk we outline that for every $1<p\le 2$ and for every continuous function $f:[0,1]\times \mathbb{R}\to \mathbb{R}$, which is Lipschitz continuous in the second variable, uniformly with respect to the first one, each bounded solution of the one-dimensional heat equation $$u_t-(|u_x{|}^{p-2}{u}_x{)}_x+f(x,u)=0\text{in}(0,1)\times (0,+\mathrm{\infty })$$ with homogeneous Dirichlet boundary conditions converges as $t\to +\mathrm{\infty }$ to a stationary solution. The proof follows an idea of Matano which is based on a comparison principle. Thus, a key step is to prove a comparison principle on non-cylindrical open sets.

Check also the PDE Seminar page. Enquiries to Florica C�rstea or Daniel Daners.