Advances in Differential Equations

Global solutions of higher-order semilinear parabolic equations in the supercritical range

Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev, and S. I. Pohozaev

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We study the asymptotic behaviour of global bounded solutions of the Cauchy problem for the semilinear $2m$-th order parabolic equation $$ u_t = -(-\Delta)^m u + |u|^p \quad {\rm in} \,\,\, {{\bf R}^N} \times \mathbb R_+, $$ where $m>1$, $p>1$, with bounded integrable initial data $u_0$. We prove that, in the supercritical Fujita range $p > p_F = 1+2m/N$, any small global solution with nonnegative initial mass, $\int u_0 dx \ge 0$, exhibits as $t \to \infty$ the asymptotic behaviour given by the fundamental solution of the linear parabolic operator (unlike the case $p \in (1,p_F]$ where solutions blow-up for any arbitrarily small nonnegative nontrivial initial data). A discrete spectrum of other possible asymptotic patterns and the corresponding monotone sequence of critical exponents $\{p_l= 1+2m/(l+N), \, l=0,1,2,...\}$, where $p_0=p_F$, are discussed.

Article information

Adv. Differential Equations Volume 9, Number 9-10 (2004), 1009-1038.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K30: Initial value problems for higher-order parabolic equations
Secondary: 35B40: Asymptotic behavior of solutions 35K55: Nonlinear parabolic equations


Egorov, Yu. V.; Galaktionov, V. A.; Kondratiev, V. A.; Pohozaev, S. I. Global solutions of higher-order semilinear parabolic equations in the supercritical range. Adv. Differential Equations 9 (2004), no. 9-10, 1009--1038.

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