The blow-up estimate for heat equations with nonlinear boundary conditions
type: Item
instance of: scholarly article
title: The blow-up estimate for heat equations with nonlinear boundary conditions
zbMATH Open document ID: 1063.35086
published in: Applied Mathematics and Computation
publication date: 2004-11-23T00:00:00Z
review text: The paper is devoted to the study of the blow-up estimates of solutions to the following problem: \[ \begin{alignedat}{3} u_t &= \Delta u, &\qquad v_t &= \Delta v &&\qquad (x \in \Omega,\;t>0),\\ \frac{\partial u}{\partial \eta} &= u^{\alpha} v^p, &\qquad \frac{\partial v}{\partial \eta} &= u^q v^{\beta} &&\qquad (x \in \partial \Omega,\;t>0),\\ u(x,0)&=u_o(x), &\qquad v(x,0) &= v_o(x) &&\qquad (x \in \Omega). \end{alignedat} \] Here \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with smooth boundary and \(\eta\) is the exterior normal vector on the boundary. It is supposed that \(\alpha, \beta \geq 0\) and \(p,q >0\) and that the initial data \(u_o, \, v_o\) are continuous nonnegative functions satisfying the compatibility conditions. The paper is closely related to [\textit{M. Pedersen} and \textit{Z. G. Lin}, J. Math. Anal. Appl. 255, 551--563 (2001; Zbl 0980.35058)]. Using different methods the author generalizes the results presented there.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 2118021
zbMATH Keywords: blow up rate
author: Wenyan Chen
reviewed by: Hanna Marcinkowska
MaRDI profile type: Publication
full work available at URL: j.amc.2003.07.015
OpenAlex ID: W1966417289
cites work: Parabolic evolution equations and nonlinear boundary conditions
cites work: Blow-up rates for parabolic systems
cites work: Q5520962
cites work: Q4272982
cites work: Remarks on the blowup estimate for solution of the heat equation with a nonlinear boundary condition
cites work: The Profile Near Blowup Time for Solution of the Heat Equation with a Nonlinear Boundary Condition
cites work: The blow-up properties of solutions to semilinear heat equations with nonlinear boundary conditions
cites work: Blow-up estimates of the positive solution of a parabolic system
cites work: The Blow-up Rate for a System of Heat Equations with Non-trivial Coupling at the Boundary
cites work: Parabolic systems with nonlinear boundary conditions
cites work: Blow-up rate estimates for semilinear parabolic systems
cites work: Blow-up estimates for semilinear parabolic systems coupled in an equation and a boundary condition.
cites work: Blow-up rates for semilinear parabolic systems with nonlinear boundary conditions
cites work: BLOW-UP PROPERTIES OF SOLUTIONS TO PARABOLIC SYSTEMS COUPLED IN EQUATIONS AND BOUNDARY CONDITIONS
cites work: Reaction diffusion systems with nonlinear boundary conditions
cites work: The blow-up rate for a system of heat equations completely coupled in the boundary conditions
DOI: 10.1016/J.AMC.2003.07.015
description: scientific article; zbMATH DE number 2118021