In this talk we outline that for every 1<p≤2 and for every continuous function f:[0,1]×ℝ→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 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.