Regularity results for non smooth parabolic problems

Abstract

In this paper we deal with the study of some regularity properties of weak solutions to non-linear, second-order parabolic equations and systems of the type \[ u_{t}-{\operatorname{div}} A(Du)=0 \;,\;\;\; (x,t)\in \Omega \times (-T,0)=\Omega_{T}, \] where $\Omega \subset {\mathbb{R}}^{n}$ is a bounded domain, $T>0$, $A:{\mathbb{R}}^{nN}\to {\mathbb{R}}^{N}$ satisfies a $p$-growth condition and $u:\Omega_{T}\to {\mathbb{R}}^{N}$. In particular, we focus our attention on local regularity of the spatial gradient of solutions of problems characterized by weak differentiability and ellipticity assumptions on the vector field $A(z)$. We prove the local Lipschitz continuity of solutions in the scalar case ($N=1$). We extend this result in some vectorial cases under an additional structure condition. Finally, we prove higher integrability and differentiability of the spatial gradient of solutions for general systems.