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

摘要

In this article we prove that for every 1 < p ≤ 2 and for every continuous function f: [0,1] × ? → ?, 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 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.