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.
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