Universal bounds for global solutions of a forced porous medium equation
type: Item
instance of: scholarly article
title: Universal bounds for global solutions of a forced porous medium equation
zbMATH Open document ID: 1065.35161
author: Michael Winkler
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-09-23T00:00:00Z
review text: The author studies the question of obtaining universal a priori estimates for global nonnegative solutions of the problem \[ \begin{aligned} u_t - \Delta\,u^m &= u^p\quad \text{in } \Omega\times (0,T),\\ \left.u\right| _{\partial\Omega} &= 0,\\ \left. u\right| _{t = 0} &= u_0 \end{aligned} \] in a smooth bounded domain \(\Omega\subset\mathbb R^n\), \(n > 1\), with \(1 < m < p\). Here \(u_0\) is a nonnegative continuous function vanishing at \(\partial\Omega\). As a result, the author proves that all global solutions to the problem are uniformly bounded in \(\Omega\times (\tau,\infty)\) by a constant depending on \(\Omega\), \(p\) and \(\tau\) but not on \(u_0\), provided that \(1 < m < p < [(n+1)/(n-1)]m\). Futhermore, it is proven the a priori bound in \(L^{\infty}(\Omega\times (0,\infty))\) depending on \(\| u_0\| _{L^{\infty}(\Omega)}\) under the optimal condition \(1 < m < p < [(n+2)/(n-2)]m\).
reviewed by: Vladimir N. Grebenev
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35B50
zbMATH DE Number: 2102511
zbMATH Keywords: global nonnegative solution
zbMATH Keywords: zero boundary data
MaRDI profile type: Publication
full work available at URL: j.na.2004.02.019
OpenAlex ID: W2040750272
cites work: A new approach to initial traces in nonlinear filtration
cites work: Stabilization of solutions of a degenerate nonlinear diffusion problem
cites work: Solutions globales d'equations de la chaleur semi lineaires.
cites work: Q4343193
cites work: Boundedness of global solutions of nonlinear diffusion equations
cites work: Linear and nonlinear heat equations in \(L^q_\delta\) spaces and universal bounds for global solutions
cites work: A bound for global solutions of semilinear heat equations
cites work: A priori bounds for positive solutions of nonlinear elliptic equations
cites work: Q5562267
cites work: On the asymptotic behavior of solutions of certain quasilinear parabolic equations
cites work: Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations
cites work: Universal bound for global positive solutions of a superlinear parabolic problem
cites work: Blow-up in quasilinear parabolic equations. Transl. from the Russian by Michael Grinfeld
cites work: A degenerate diffusion equation with a nonlinear source term
DOI: 10.1016/J.NA.2004.02.019
description: scientific article; zbMATH DE number 2102511