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

In this article we prove that for every ${1 < p le 2}$ and for every continuous function ${f: [0,1]timesmathbb{R}tomathbb{R}}$ , which is Lipschitz continuous in the second variable, uniformly with respect to the first one, each bounded solution of the one-dimensional heat equation $$begin{array}{ll}u_{t}-{|u_{x}|^{p-2}u_{x} }_{x}+f(x,u)=0qquad{rm in} quad (0,1)times (0,+infty) end{array}$$ with homogeneous Dirichlet boundary conditions converges as ${tto+infty}$ 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.