## Advances in Differential Equations

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

### 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

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867912

**Mathematical Reviews number (MathSciNet)**

MR2098064

**Zentralblatt MATH identifier**

1122.35040

**Subjects**

Primary: 35K30: Initial value problems for higher-order parabolic equations

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

#### Citation

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. https://projecteuclid.org/euclid.ade/1355867912