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 u_t 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 u_xi belong to L_∞ as well. In the singular case we show that the second derivatives u_xixj of a solution of the Cauchy problem belong to L₂.