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

Abstract

In this talk we outline that for every $1<p\le 2$ and for every continuous function $f:\left[0,1\right]\times ℝ\to ℝ$, which is Lipschitz continuous in the second variable, uniformly with respect to the first one, each bounded solution of the one-dimensional heat equation

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.