Renormalized solutions of nonlinear parabolic equations with general measure data

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.