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 \[ \left\{ \begin{array} [c]{l}% {u_{t}}-{Δ_{p}}u=μ\qquad\text{in }Q,\\ {u}=0\qquad\text{on }\partialΩ\times(0,T),\\ u(0)=u_{0}\qquad\text{in }Ω, \end{array} \right. \] 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{ } \[ \left\{ \begin{array} [c]{l}% {u_{t}}-{Δ_{p}}u+\mathcal{G}(u)=μ\qquad\text{in }Q,\\ {u}=0\qquad\text{on }\partialΩ\times(0,T),\\ u(0)=u_{0}\qquad\text{in }Ω, \end{array} \right. \] where $\mathcal{G}(u)$ may be an absorption or a source term$.$ In the model case $\mathcal{G}(u)=\pm {\left|u\right|}^{q-1}u$ $(q>p-1),$ 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.

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• Parabolic equations with exponential nonlinearity and measure data