Critical global asymptotics in higher-order semilinear parabolic equations

Abstract

We consider a higher-order semilinear parabolic equationut=−(−Δ)mu−g(x,u)inℝN×ℝ+,m>1. The nonlinear term is homogeneous:g(x,su)≡|s|p−1sg(x,u)andg(sx,u)≡|s|Qg(x,u)for anys∈ℝ, with exponentsP>1, andQ>−2m. We also assume thatgsatisfies 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 exponentP=1+(2m+Q)/Nsuch that the asymptotic behavior ast→∞of a class of global small solutions is not group-invariant and is given by a logarithmic perturbation of the fundamental solutionb(x,t)=t−N/2mf(xt−1/2m)of the parabolic operator∂/∂t+(−Δ)m, so that fort≫1,u(x,t)=C0(ln t)−N/(2m+Q)[b(x,t)+o(1)], whereC0is a constant depending onm,N, andQonly.