Stability properties for quasilinear parabolic equations with measure data

Let $\mathrm{\Omega }$ be a bounded domain of ${\mathbb{R}}^N$, and $Q=\Omega \times(0,T).$ We study problems of the model type \left\{ \begin{array} [c]{l}% {u_{t}}-{\Delta_{p}}u=\mu\qquad\text{in }Q,\\ {u}=0\qquad\text{on }\partial\Omega\times(0,T),\\ u(0)=u_{0}\qquad\text{in }\Omega, \end{array} \right. where $p>1$, $\mu \in {\mathcal{M}}_b(Q)$ and $u_{0}\in L^{1}(\Omega).$ Our main result is a $\mathit{\text{stability}}\mathit{\text{theorem}}$extending the results of Dal Maso, Murat, Orsina, Prignet, for the elliptic case, valid for quasilinear operators $u⟼\mathcal{A}(u)=$div$(A(x,t,\nabla u))$$\mathit{\text{.}}$