Renormalized solutions of nonlinear parabolic equations with general measure data

TL;DR

This paper establishes the existence of renormalized solutions for nonlinear parabolic equations with measure data, extending the theory to cases with general measure data and $p$-Laplacian operators.

Contribution

It proves the existence of solutions for nonlinear parabolic equations with measure data, broadening the scope of previous results to more general measures and boundary conditions.

Findings

Existence of renormalized solutions for measure data

Applicable to general bounded measures in space-time

Handles nonlinear p-Laplacian operators

Abstract

Let $\Omega\subseteq \mathbb{R}^N$ a bounded open set, $N\geq 2$, and let $p>1$; we prove existence of a renormalized solution for parabolic problems whose model is $$ \begin{cases} u_{t}-\Delta_{p} u=\mu & \text{in}\ (0,T)\times\Omega,\newline u(0,x)=u_0 & \text{in}\ \Omega,\newline u(t,x)=0 &\text{on}\ (0,T)\times\partial\Omega, \end{cases} $$ where $T>0$ is any positive constant, $\mu\in M(Q)$ is a any measure with bounded variation over $Q=(0,T)\times\Omega$, and $u_o\in L^1(\Omega)$, and $-\Delta_{p} u=-{\rm div} (|\nabla u|^{p-2}\nabla u )$ is the usual $p$-laplacian.