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On a general approach to extinction and blow-up for quasi-linear heat equations

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Velázquez, J.J. L. and Galaktionov, V. A. and Posashkov, S. A. and Herrero, Miguel A. (1993) On a general approach to extinction and blow-up for quasi-linear heat equations. Computational Mathematics and Mathematical Physics, 33 (2). pp. 217-227. ISSN 0965-5425

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Official URL: http://www.mathnet.ru/links/d7d26db03f5293989ec731fcacfa35d5/zvmmf2764.pdf




Abstract

The authors study asymptotic behaviour of positive solutions of equations of the type ut =Δφ(u)±Q(u), where φ′ and Q are given positive functions. By determining an auxiliary function F(u) appearing in an expression posed by A. Friedman and B. McLeod, they obtain asymptotic estimates of solutions as t→T, blow-up or extinction time. These estimates have been established by other authors using different methods. Moreover, the paper poses a conjecture that, if the behaviour of u(0,t) as t→T near a blow-up or extinction point is known, all the information about the corresponding asymptotic expansions on small compact subsets near the origin is encoded in the first order ODE φ′(u)ur+rF(u)=0 for r>0 as t→T, where an optimal choice of F(u) is indicated in the paper.


Item Type:Article
Uncontrolled Keywords:Friedman-McLeod method; blow-up; extinction; point; heat equation
Subjects:Sciences > Mathematics > Differential equations
ID Code:18007
Deposited On:29 Jan 2013 09:44
Last Modified:19 Feb 2019 13:08

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