Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data
Alberto Fiorenza Anna Mercaldo Jean Michel Rakotoson
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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