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 \(1<p\leq 2\) and for every continuous function \(f\colon [0,1]\times\mathbb R\to\mathbb R\), which is Lipschitz continuous in the second variable, uniformly with respect to the first one, each bounded solution of the one-dimensional heat equation \[ u_{t}-\bigl(|u_{x}|^{p-2}u_{x}\bigr)_{x}+f(x,u)=0 \qquad\text{in}\quad (0,1)\times (0,+\infty) \] with homogeneous Dirichlet boundary conditions converges as \(t\to +\infty\) 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 Cîrstea or Daniel Daners.