Differential and Integral Equations

Examples of nonsymmetric extinction and blow-up for quasilinear heat equations

Victor A. Galaktionov and Sergey A. Posashkov

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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.

Article information

Differential Integral Equations, Volume 8, Number 1 (1995), 87-103.

First available in Project Euclid: 21 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B40: Asymptotic behavior of solutions


Galaktionov, Victor A.; Posashkov, Sergey A. Examples of nonsymmetric extinction and blow-up for quasilinear heat equations. Differential Integral Equations 8 (1995), no. 1, 87--103. https://projecteuclid.org/euclid.die/1369143785

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