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

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.