Lipschitz continuous solutions of some doubly nonlinear parabolic equations

Abstract

This paper is concerned with two types of nonlinear parabolic equations, which arise from the nonlinear filtration problems for non-Newtonian fluids. These equations include as special cases the porous medium equations $u_t =$ div$(u^l\nabla u)$ and the evolution equation governed by p-Laplacian $u_t =$ div $(|\nabla u|^{p-2}\nabla u)$. Because of the degeneracy or singularity caused by the terms $u^l$ and $|\nabla u|^{p-2}$, one can not expect the existence of global (in time) classical solutions for these equations except for special cases. Therefore most of works have been devoted to the study of weak solutions. The main purpose of this paper is to investigate the existence of much more regular (not necessarily global) solutions. The existence of local solutions in $W^{1,\infty}(\Omega)$ is assured under the assumption that initial data are non-negative functions in $W_0^{1,\infty}(\Omega)$, and that the mean curvature of the boundary $\partial \Omega$ of the domain $\Omega$ is non-positive. We here introduce a new method $L^\infty$-energy method, which provides a main tool for our arguments and would be useful for other situations.