# American Institute of Mathematical Sciences

November  2002, 8(4): 893-906. doi: 10.3934/dcds.2002.8.893

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

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

Received  January 2001 Revised  March 2002 Published  July 2002

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.

Citation: Alberto Fiorenza, Anna Mercaldo, Jean Michel Rakotoson. Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 893-906. doi: 10.3934/dcds.2002.8.893
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