Existence of Lipschitz continuous solutions to the Cauchy–Dirichlet problem for anisotropic parabolic equations

Abstract

The Cauchy–Dirichlet and the Cauchy problem for the degenerate and singular quasilinear anisotropic parabolic equations are considered. We show that the time derivative ut of a solution u belongs to L∞ under a suitable assumption on the smoothness of the initial data. Moreover, if the domain satisfies some additional geometric restrictions, then the spatial derivatives uxi belong to L∞ as well. In the singular case we show that the second derivatives uxixj of a solution of the Cauchy problem belong to L2.