Boundedness for some doubly nonlinear parabolic equations in measure spaces
type: Item
instance of: scholarly article
title: Boundedness for some doubly nonlinear parabolic equations in measure spaces
zbMATH Open document ID: 1401.35164
author: Eurica Henriques
author: Rojbin Laleoglu
published in: Journal of Dynamics and Differential Equations
publication date: 2018-11-06T00:00:00Z
review text: The authors derive interesting local and global sup bounds of the nonnegative weak subsolutions of \[ (u^q)_t-\nabla\cdot(|\nabla u|^{p-2}\nabla u)=0, \] in \(U_\tau=U\times(\tau_1,\tau_2],\quad p>1,\quad q>1\) and of its associated Dirichlet problem. All these results are obtained in measure spaces equipped with a doubling non-trivial Borel measure supporting a Poincaré inequality. For particular ranges of the exponents \(p\) and \(q\), the authors show that any locally nonnegative weak subsolution, taken in \(Q\subset\bar Q\subset U\tau\), is controlled from above by the \(L^\alpha(\bar Q)\)-norm, for \(\alpha=\max\{p,q+1\}\). As for the global setting, under suitable assumptions on the boundary datum \(g\) and on the initial datum, the authors obtain sup bounds for \(u\), in \(U\times\{t\}\), which depend on the sup of \(g\) and on the \(L^{q+1}(U\times(\tau_1,\tau_1+t])\)-norm of \((u-\sup g)_+\), for all \(t\in (0,\tau_2-\tau_1]\). On the critical ranges of \(p\) and \(q\), as expected, a priori local and global \(L^\infty\) estimates require extra qualitative information on \(u\).
reviewed by: Vincenzo Vespri
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35K67
zbMATH DE Number: 6973926
zbMATH Keywords: boundedness
zbMATH Keywords: singular PDE
zbMATH Keywords: degenerate PDE
zbMATH Keywords: doubly nonlinear PDE
MaRDI profile type: Publication
full work available at URL: s10884-017-9585-3
OpenAlex ID: W2604015075
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DOI: 10.1007/S10884-017-9585-3
description: scientific article; zbMATH DE number 6973926