Note on critical exponents for a system of heat equations coupled in the boundary conditions
type: Item
instance of: scholarly article
title: Note on critical exponents for a system of heat equations coupled in the boundary conditions
zbMATH Open document ID: 0891.35052
author: Shu Wang
author: Chunhong Xie
author: Mingxin Wang
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-07-01T00:00:00Z
review text: The authors study the blow-up rate and uniqueness of nonnegative solutions of the system of heat equations \(u_t=u_{xx}\), \(v_t=v_{xx}\) on the half-line \(x>0\), coupled by the nonlinear boundary conditions \(u_x(0,t)=-v^p(0,t)\), \(v_x(0,t)=-u^q(0,t)\), where \(p,q>0\). \textit{K. Deng, M. Fila}, and \textit{H. A. Levine} [Acta Math. Univ. Comen., New Ser. 63, No. 2, 169-192 (1994; Zbl 0824.35048)] showed nonuniqueness of solutions of this system if \(pq<1\) and blow-up in finite time if \(pq>1\). In the latter case, they also established the corresponding blow-up rate for some class of initial conditions provided \(\min\{p,q\}\geq1\). The authors of the paper under review generalize the results on the blow-up rate by enlarging the class of admissible initial conditions and by removing the assumption \(\min\{p,q\}\geq1\). Moreover, they show uniqueness of solutions (of a higher dimensional analogue) of the system provided \(pq\geq1\).
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 1123527
zbMATH Keywords: blow-up rate
zbMATH Keywords: uniqueness
zbMATH Keywords: nonlinear boundary condition
reviewed by: Pavol Quittner
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5751
OpenAlex ID: W2069550553
cites work: Parabolic evolution equations and nonlinear boundary conditions
cites work: Q4764290
cites work: Boundedness and blow up for a semilinear reaction-diffusion system
cites work: A Uniqueness Result for a Semilinear Reaction-Diffusion System
cites work: Q4880668
cites work: The blow‐up rate for the heat equation with a non‐linear boundary condition
cites work: Q5520962
cites work: On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary
cites work: Asymptotically self‐similar blow‐up of semilinear heat equations
cites work: Q3736159
cites work: The Profile Near Blowup Time for Solution of the Heat Equation with a Nonlinear Boundary Condition
cites work: Q5562267
cites work: The Role of Critical Exponents in Blowup Theorems
cites work: The value of the critical exponent for reaction-diffusion equations in cones
cites work: Nonlinear elliptic boundary-value problems in unbounded domains
cites work: Q4023219
DOI: 10.1006/JMAA.1997.5751
description: scientific article; zbMATH DE number 1123527