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

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.