Asymptotics for critical nonconvective type equations

Nakao Hayashi,Pavel I Naumkin,Elena I Kaikina

doi:10.1155/s0161171204303133

Abstract

We study large‐time asymptotic behavior of solutions to the Cauchy problem for a model of nonlinear dissipative evolution equation. The linear part is a pseudodifferential operator and the nonlinearity is a cubic pseudodifferential operator defined by means of the inverse Fourier transformation and represented by bilinear and trilinear forms with respect to the direct Fourier transform of the dependent variable. We consider nonconvective type nonlinearity, that is, we suppose that the total mass of the nonlinear term does not vanish. We consider the initial data, which have a nonzero total mass and belong to the weighted Sobolev space with a sufficiently small norm. Then we give the main term of the large‐time asymptotics of solutions in the critical case. The time decay rate have an additional logarithmic correction in comparison with the corresponding linear case.

Highlights

ASYMPTOTICS FOR CRITICAL NONCONVECTIVE TYPE EQUATIONSWe study large-time asymptotic behavior of solutions to the Cauchy problem for a model of nonlinear dissipative evolution equation
We study large-time asymptotics of solutions to the Cauchy problem for dissipative equations ut + ᏺ(u) + ᏸu = 0, t > 0, u(0, x) = u0(x), x ∈ R
The linear part of (1.1) is a pseudodifferential operator defined by the Fourier transformation ᏸu = Ᏺξ→x L(ξ)u(ξ), (1.2)

Summary

ASYMPTOTICS FOR CRITICAL NONCONVECTIVE TYPE EQUATIONS

We study large-time asymptotic behavior of solutions to the Cauchy problem for a model of nonlinear dissipative evolution equation. The linear part is a pseudodifferential operator and the nonlinearity is a cubic pseudodifferential operator defined by means of the inverse Fourier transformation and represented by bilinear and trilinear forms with respect to the direct Fourier transform of the dependent variable. We consider nonconvective type nonlinearity, that is, we suppose that the total mass of the nonlinear term does not vanish. We consider the initial data, which have a nonzero total mass and belong to the weighted Sobolev space with a sufficiently small norm. We give the main term of the large-time asymptotics of solutions in the critical case. The time decay rate have an additional logarithmic correction in comparison with the corresponding linear case.

Introduction

Denote t