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

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