, Volume 187, Issue 4, pp 563-604

Renormalized solutions of nonlinear parabolic equations with general measure data

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Abstract

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