Stability properties for quasilinear parabolic equations with measure data

TL;DR

This paper establishes a stability theorem for quasilinear parabolic equations with measure data, extending previous elliptic case results to time-dependent problems involving divergence form operators.

Contribution

It introduces a stability result for solutions of quasilinear parabolic equations with measure data, generalizing known elliptic case theorems to the parabolic setting.

Findings

Proves stability of solutions under measure data approximation.

Extends elliptic stability results to parabolic equations.

Provides a framework for analyzing measure data in quasilinear parabolic problems.

Abstract

Let $\Omega$ 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 \textit{stability theorem }extending the results of Dal Maso, Murat, Orsina, Prignet, for the elliptic case, valid for quasilinear operators $u\longmapsto\mathcal{A}(u)=$div$(A(x,t,\nabla u))$\textit{. }