Advances in Differential Equations

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

F. Crispo and P. Maremonti

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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

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

First available in Project Euclid: 2 July 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B65: Smoothness and regularity of solutions 35K51: Initial-boundary value problems for second-order parabolic systems 35K67: Singular parabolic equations


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.

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