Journal d’Analyse Mathématique

, Volume 90, Issue 1, pp 141–173

Global existence and time decay of small solutions to the landau-ginzburg type equations

• Nakao Hayashi
• Elena I. Kaikina
• Pavel I. Naumkin
Article

Abstract

We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|2/n u = 0,x ∃ Rn,t } 0,u(0,x) = u0(x),x ∃ Rn, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β) 0, θ = ¦∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|n-1nn/2 / ((n + 1)|α|2 + α2 n/2. Furthermore, we assume that the initial data u0 ∃ Lp are such that (1 + ¦x¦)αu0 ∃ L1, with sufficiently small norm ∃ = (1 + ¦x¦)α u0 1 + u0 p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L) ∩ C ([0, ∞); L1 ∩ Lp) satisfying the time decay estimates for allt0 u(t)|| Cɛt-n/2(1 + η log 〈t〉)-n/2, if hg = θ2/n 2π ℜδ(α, Β) 0; u(t)|| Cɛt-n/2(1 + Μ log 〈t〉)-n/4, if η = 0 and Μ = θ4/n 4π)2 (ℑδ(α, Β))2 ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)|| Cɛt-n/2(1 + κ log 〈t〉)-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31).

Keywords

Cauchy Problem Remainder Term Small Solution Large Time Behavior Taylor Formula

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© Hebrew University of Jerusalem 2003

Authors and Affiliations

• Nakao Hayashi
• 1
• Elena I. Kaikina
• 2
• Pavel I. Naumkin
• 3
1. 1.Department of Mathematics Graduate School of ScienceOsaka UniversityToyonaka OsakaJapan
2. 2.Departamento de Ciencias BásicasInstituto tecnológico de moreliaMoreliaMéxico
3. 3.Instituto de MatemáticasUNAMMoreliaMéxico