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

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.