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2004 Global solutions of higher-order semilinear parabolic equations in the supercritical range
Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev, S. I. Pohozaev
Adv. Differential Equations 9(9-10): 1009-1038 (2004).

Abstract

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.

Citation

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Yu. V. Egorov. V. A. Galaktionov. V. A. Kondratiev. S. I. Pohozaev. "Global solutions of higher-order semilinear parabolic equations in the supercritical range." Adv. Differential Equations 9 (9-10) 1009 - 1038, 2004.

Information

Published: 2004
First available in Project Euclid: 18 December 2012

zbMATH: 1122.35040
MathSciNet: MR2098064

Subjects:
Primary: 35K30
Secondary: 35B40, 35K55

Rights: Copyright © 2004 Khayyam Publishing, Inc.

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Vol.9 • No. 9-10 • 2004
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