### Higher regularity of solutions to the singular $p$-Laplacean parabolic system

#### Abstract

We study existence and regularity properties of solutions to the singular $p$-Laplacian parabolic system in a bounded domain $\Omega$. The main purpose is to prove global $L^r(\varepsilon,T;L^q(\Omega))$, $\varepsilon\geq0$, integrability properties of the second spatial derivatives and of the time derivative of the solutions. Hence, for suitable $p$ and exponents $r$ and $q$, by Sobolev embedding theorems, we deduce global regularity of $u$ and $\nabla u$ in Hölder spaces. Finally we prove a global pointwise bound for the solution under the assumption $p>\frac{2n}{n+2}$.

#### Article information

Source
Adv. Differential Equations Volume 18, Number 9/10 (2013), 849-894.

Dates
First available in Project Euclid: 2 July 2013

Crispo, F.; Maremonti, P. Higher regularity of solutions to the singular $p$-Laplacean parabolic system. Adv. Differential Equations 18 (2013), no. 9/10, 849--894. https://projecteuclid.org/euclid.ade/1372777762