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1995 Examples of nonsymmetric extinction and blow-up for quasilinear heat equations
Victor A. Galaktionov, Sergey A. Posashkov
Differential Integral Equations 8(1): 87-103 (1995).

Abstract

We present new asymptotic properties of solutions of some quasilinear heat equations with absorption or source including the equations $$ u_t = \nabla \cdot (u^{1/2} \nabla u) - u^{1/2}, \quad u_t = \nabla \cdot (u^{-1/2} \nabla u) - u^{1/2}, $$ $$ u_t = \nabla \cdot (u^{-1/2} \nabla u) + u^{3/2}, $$ $$ u_t = \nabla \cdot (u^{-4/(N+2)} \nabla u) + u^{(N+6)/(N+2)}, \quad x \in \Bbb R^N , \,\, N \ge 1. $$ We show that these equations admit explicit solutions which are nonsymmetric and nonmonotone in spatial variables near the extinction of blow-up time. The corresponding equations are shown to be reduced to finite dimensional dynamical systems on linear subspaces which are invariant under certain nonlinear reaction-diffusion operators.

Citation

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Victor A. Galaktionov. Sergey A. Posashkov. "Examples of nonsymmetric extinction and blow-up for quasilinear heat equations." Differential Integral Equations 8 (1) 87 - 103, 1995.

Information

Published: 1995
First available in Project Euclid: 21 May 2013

zbMATH: 0814.35047
MathSciNet: MR1296111

Subjects:
Primary: 35K55
Secondary: 35B40

Rights: Copyright © 1995 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.8 • No. 1 • 1995
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