### 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.

#### Article information

Source
Adv. Differential Equations, Volume 13, Number 3-4 (2008), 367-398.

Dates
First available in Project Euclid: 18 December 2012

Mathematical Reviews number (MathSciNet)
MR2482422

Zentralblatt MATH identifier
1160.35019

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B65: Smoothness and regularity of solutions 35D10

#### Citation

Pisante, Giovanni; Verde, Anna. Regularity results for non smooth parabolic problems. Adv. Differential Equations 13 (2008), no. 3-4, 367--398. https://projecteuclid.org/euclid.ade/1355867354