International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2003 / Article

Open Access

Volume 2003 |Article ID 406190 | https://doi.org/10.1155/S0161171203210176

Victor A. Galaktionov, "Critical global asymptotics in higher-order semilinear parabolic equations", International Journal of Mathematics and Mathematical Sciences, vol. 2003, Article ID 406190, 17 pages, 2003. https://doi.org/10.1155/S0161171203210176

Critical global asymptotics in higher-order semilinear parabolic equations

Received18 Oct 2002

Abstract

We consider a higher-order semilinear parabolic equation ut=(Δ)mug(x,u) in N×+, m>1. The nonlinear term is homogeneous: g(x,su)|s|p1sg(x,u) and g(sx,u)|s|Qg(x,u) for any s, 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)=tN/2mf(xt1/2m) of the parabolic operator /t+(Δ)m, so that for t1, u(x,t)=C0(lnt)N/(2m+Q)[b(x,t)+o(1)], where C0 is a constant depending on m, N, and Q only.

Copyright © 2003 Hindawi Publishing Corporation. 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.


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