Renormalized solutions of nonlinear parabolic equations with general measure data

Abstract

Let $\Omega \subseteq {\mathbb{R}}^N$ a bounded open set, $N\ge 2$, and let $p>1$; we prove existence of a renormalized solution for parabolic problems whose model is $$\{\begin{array}{ll}{u}_t-{\Delta }_{p}u=\mu & \text{in}(0,T)\times \Omega ,\\ u(0,x)=u_0& \text{in}\Omega ,\\ u(t,x)=0& \text{on}(0,T)\times \partial \Omega ,\end{array}$$ 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 $-{\Delta }_{p}u=-\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)$ is the usual $p$-laplacian.

References in corpus (2)

• Inverse Problem for a Curved Quantum Guide
• Asymptotic behavior of solutions for linear parabolic equations with general measure data

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• Local estimates for parabolic equations with nonlinear gradient terms
• Obstacle problem for semilinear parabolic equations with measure data
• Global estimates for nonlinear parabolic equations
• Regularity estimates for singular parabolic measure data problems with sharp growth
• Global regularity for degenerate/singular parabolic equations involving measure data
• Smooth measures and capacities associated with nonlocal parabolic operators
• Characterizations of interior polar sets for the degenerate p-parabolic equation
• Marcinkiewicz regularity for singular parabolic $p$-Laplace type equations with measure data