Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data

Pages: 893 - 906, Volume 8, Issue 4, October 2002      doi:10.3934/dcds.2002.8.893

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Alberto Fiorenza - Dipartimento di Costruzioni e Metodi Matematici in Architettura, Universitá di Napoli "Federico II", via Monteoliveto, 3, I-80134 Napoli, Italy (email)
Anna Mercaldo - Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Universitá di Napoli "Federico II", via Cintia, I-80126 Napoli, Italy (email)
Jean Michel Rakotoson - Laboratoire d'Applications des Mathématiques, Teleport 2 Département de Mathématiques, Université de Poitiers, B.P. 30179, 86962 Futuroscope Chasseneuil cedex, France (email)

Abstract: In this paper we prove some regularity and uniqueness results for a class of nonlinear parabolic problems whose prototype is

$\partial_t u - \Delta_N u=\mu$ in $\mathcal D'(Q) $

$u=0$ on $]0,T[\times\partial \Omega$

$u(0)=u_0$ in $ \Omega,$

where $Q$ is the cylinder $Q=(0,T)\times\Omega$, $T>0$, $\Omega\subset \mathbb R^n$, $N\ge 2$, is an open bounded set having $C^2$ boundary, $\mu\in L^1(0,T;M(\Omega))$ and $u_0$ belongs to $M(\Omega)$, the space of the Radon measures in $\Omega$, or to $L^1(\Omega)$. The results are obtained in the framework of the so-called grand Sobolev spaces, and represent an extension of earlier results on standard Sobolev spaces.

Keywords:  Grand Sobolev spaces, measure data, uniqueness, regularity, parabolic equations.
Mathematics Subject Classification:  35K60, 35R05, 35K15, 46E30.

Received: January 2001;      Revised: March 2002;      Available Online: July 2002.