Stability properties for quasilinear parabolic equations with measure data and applications

Abstract: Let $\Omega$ be a bounded domain of $\mathbb{R}^{N}$, and $Q=\Omega \times(0,T).$ We first study the problem [ \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 $p>1$, $\mu\in\mathcal{M}{b}(\Omega)$ and $u{0}\in L^{1}(\Omega).$ Our main result is a \textit{stability theorem }extending the results of Dal Maso, Murat, Orsina, Prignet, for the elliptic case\textit{. } As an application, we consider the perturbed problem\textit{ } [ \left{ \begin{array} [c]{l}% {u_{t}}-{\Delta_{p}}u+\mathcal{G}(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 $\mathcal{G}(u)$ may be an absorption or a source term$.$ In the model case $\mathcal{G}(u)=\pm\left\vert u\right\vert ^{q-1}u$ $(q>p-1),$ or $\mathcal{G}$ has an exponential type. We give existence results when $q$ is subcritical, or when the measure $\mu$ is good in time and satisfies suitable capacity conditions.