Critical global asymptotics in higher-order semilinear parabolic equations

Victor A Galaktionov

doi:10.1155/s0161171203210176

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.

Highlights

The basic example is a 2mth-order semilinear parabolic equation in the critical case, where a special nonlinear interaction between operators produces asymptotics perturbed by logarithmic factors
In order to guarantee the existence of global solutions, without loss of generality of the asymptotic technique to be applied, we assume that the perturbation term g satisfies a coercivity condition to ensure the existence of a local solution of the integral equation obtained by means of application of the continuous semigroup generated by −(−∆)m
We introduce the crucial assumption on the critical nonlinearity ensuring the existence of special noninvariant global asymptotics

Summary

Introduction

The basic example is a 2mth-order semilinear parabolic equation in the critical case, where a special nonlinear interaction between operators produces asymptotics perturbed by logarithmic factors. We will show that in the critical case P = Pc, under certain assumptions of g and initial data, there exist small global solutions with the following asymptotic behavior as t → ∞: u(x, t) = ±C0t−N/2m(ln t)−N/(2m+Q) f x t1/2m

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Conclusion