International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 60, Pages 3809-3825
doi:10.1155/S0161171203210176

Critical global asymptotics in higher-order semilinear parabolic equations

Victor A. Galaktionov1,2

1Keldysh Institute of Applied Mathematics, Miusskaya Square 4, Moscow 125047, Russia
2Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK

Received 18 October 2002

Copyright © 2003 Victor A. Galaktionov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a higher-order semilinear parabolic equation ut=−(−Δ)mu−g(x,u) in ℝN×ℝ+, m>1. The nonlinear term is homogeneous: g(x,su)≡|s|p−1sg(x,u) and g(sx,u)≡|s|Qg(x,u) for any s∈ℝ, with exponents P>1, and Q>−2m. We also assume that g satisfies necessary coercivity and monotonicity conditions for global existence of solutions with sufficiently small initial data. The equation is invariant under a group of scaling transformations. We show that there exists a critical exponent P=1+(2m+Q)/N such that the asymptotic behavior as t→∞ of a class of global small solutions is not group-invariant and is given by a logarithmic perturbation of the fundamental solution b(x,t)=t−N/2mf(xt−1/2m) of the parabolic operator ∂/∂t+(−Δ)m, so that for t≫1, u(x,t)=C0(ln t)−N/(2m+Q)[b(x,t)+o(1)], where C0 is a constant depending on m, N, and Q only.