Bounds as \(t\to +\infty\) for solutions of degenerate linear and nonlinear parabolic equations in unbounded domains
type: Item
instance of: scholarly article
title: Bounds as \(t\to +\infty\) for solutions of degenerate linear and nonlinear parabolic equations in unbounded domains
zbMATH Open document ID: 0594.35007
DOI: 10.1007/BF01157681
author: R. Ya. Glagoleva
published in: Mathematical Notes
publication date: 1985-01-01T00:00:00Z
review text: The author gives bounds as \(t\to \infty\) of the maximum of the positive solutions of the problem \[ (1)\quad \sum^{n}_{i,k=1}(\partial /\partial x_ i)(a_{ik}(t,x,u,D_ xu)\partial u/\partial x_ k)+f(t,x,u,D_ xu)-\partial u/\partial t=0, \] \[ (2)\quad u|_ E=0,\quad \partial u/\partial \nu |_{S_ 0^{\infty}\setminus E}=0,\quad u|_{t=0}=\phi (x)\in L_ 2(G), \] where \(D_ xu=(u_{x_ 1},...,u_{x_ n})\), G is bounded or unbounded domain in \(R^ n\), \(\partial /\partial \nu\) the conormal derivative, \(S_ 0^{\infty}=\partial G\times [0,\infty)\), E a closed set in \(S_ 0^{\infty}\); \(a_{ik}\) and f satisfy some additional conditions. Together with (1), (2) the problem analogical to (2) in \(G\times [t_ 0,T]\) for the differential inequality \[ \sum^{n}_{i,k=1}(\partial /\partial x_ i)(a_{ik}(t,x)\partial u/\partial x_ k)-\partial u/\partial t\geq 0 \] is considered.
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K99
zbMATH DE Number: 3955505
zbMATH Keywords: parabolic equation
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: positive solutions
zbMATH Keywords: differential inequality
MaRDI profile type: Publication
cites work: A problem with mixed boundary conditions for a quasilinear parabolic equation
cites work: Q4199613
cites work: A harnack inequality for parabolic differential equations
full work available at URL: https://doi.org/10.1007/bf01157681
OpenAlex ID: W2043120093
description: scientific article; zbMATH DE number 3955505