Pointwise estimates via parabolic potentials for a class of doubly nonlinear parabolic equations with measure data
type: Item
instance of: scholarly article
title: Pointwise estimates via parabolic potentials for a class of doubly nonlinear parabolic equations with measure data
zbMATH Open document ID: 1401.35169
author: Stefan Sturm
published in: Manuscripta Mathematica
publication date: 2018-10-01T00:00:00Z
review text: In the space-time cylinder \(E_T:=E\times (0,T)\), where \(E\subset\mathbb R^n\) is an open bounded set, \(n\geq 2\), \(T>0\), the author considers the following Coauchy-Dirichlet problem \[ \begin{cases} u_t-\operatorname{div}\big(\mathbf A(x,t,u,Du)\big)=\mu, \quad \text{in}\quad E_T,\\ u=0,\quad \text{on}\quad \Gamma_T, \end{cases}\tag{1} \] where \(\mu\) is a non-negative Radon measure on \(E_T\) with finite total mass \(\mu(E_T)<\infty\) and \(\Gamma_T:=\big(\overline E\times \{0\}\big)\bigcup\big(\partial E\times(0,T)\big)\). The nonlinear parabolic equation \[ u_t-\operatorname{div}\big(| u|^{m-1}\,| Du|^{p-2}\,Du\big)=\mu, \quad \text{in}\quad E_T, \] is the model example for the equation from (1). The operator \(\mathbf A:E_T\times\mathbb R\times\mathbb R^n\mapsto\mathbb R\) satisfy the following ellipticity conditions: \textbf{(A1)} \(\mathbf A\) is measurable with respect to \((x,t)\in E_T\) for all \((u,\xi)\in \mathbb R\times\mathbb R^n\) and continuous with respect to \((u,\xi)\in \mathbb R\times\mathbb R^n\) for almost every \((x,t)\in E_T\);\,\, \textbf{(A2)}\, \(\big(\mathbf A(x,t,u,\xi),\xi\big)\geq C_0\,| u|^{m-1}\,|\xi|^{p}\);\,\, \textbf{(A3)}\, \(|\mathbf A(x,t,u,\xi)|\leq C_1 |u|^{m-1}\,|\xi|^{p-1}\), where \(C_0>0\), \(C_1>0\), \(p\geq 2\) and \(m>1\) in both conditions \textbf{(A2)} and \textbf{(A3)}. The main result of the paper is the following. Suppose that \(\mathbf A\) satisfies conditions \textbf{(A1)}--\textbf{(A3)} and \(\mu\in L^{q_1,q_2}_{\text{loc}}\big(E; L^{q_1,\infty}_{\text{loc}}(0,T)\big)\), for \(q_1=\frac{n+p}{p}\) and \(q_2=\frac{n+p}{n(p-1)+p}\). Then any weak solution \(u\) to the problem (1) is locally uniformly bounded in \(E_T\). The author also defines the nonlinear parabolic potentials for the equation (1) and in terms of these potentials establishes the pointwise estimates for weak solutions to the problem (1).
reviewed by: Andrei Perjan
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K20
zbMATH DE Number: 6945635
zbMATH Keywords: nonlinear parabolic equation
zbMATH Keywords: degenerate parabolic equation
zbMATH Keywords: weak solution
zbMATH Keywords: poinwise estimates
MaRDI profile type: Publication
full work available at URL: s00229-018-1014-3
OpenAlex ID: W2792098362
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Wikidata QID: Q130185298
DOI: 10.1007/S00229-018-1014-3
description: scientific article; zbMATH DE number 6945635