Global existence and time decay of small solutions to the Landau-Ginzburg type equations

Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin

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18 Citations (Scopus)

Abstract

We study the Cauchy problem for the nonlinear dissipative equations (0.1) { ∂tu - αΔu + β|u|2/nu = 0, x ∈ Rn, t > 0, u(0, x) = u0(x), x ∈ R n, 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) = β|α|n-1n n/2/((n+1)|α|2 + α2) n/2. Furthermore, we assume that the initial data u0 ∈ Lp are such that (1 + |x|)a u0 ∈ L1, with sufficiently small norm ε = ∥(1 + |x|)a u01 + ∥u0p, where p > 1, a ∈ (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 all t > 0 ∥u(t)∥ ≤ Cεt -n/2 (1 + η log 〈t〉)-n/2, if η = θ2/n/2πnℜδ(α, β) > 0; ∥u(t)∥ ≤ Cεt-n/2 (1 + μ log 〈t〉)-n/4, if η = 0 and μ = θ 4/n/(4π)2 (F-fraktur signδ(α, β)) 2 ℜ ((1 + 1/n) v1 - 1/n v2) > 0; and ∥u(t)∥ ≤ Cεt-n/2 (1 + χ log 〈t〉)-n/6, if η = 0, μ = 0, χ > 0, where vl, l = 1, 2 are defined in (1.2), x is a positive constant defined in (2.31).

Original languageEnglish
Pages (from-to)141-173
Number of pages33
JournalJournal d'Analyse Mathematique
Volume90
DOIs
Publication statusPublished - 2003 Jan 1
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)

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