Journal d’Analyse Mathématique

, Volume 90, Issue 1, pp 141–173 | Cite as

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

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


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).


Cauchy Problem Remainder Term Small Solution Large Time Behavior Taylor Formula 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Escobedo and O. Kavian,Asymptotic behavior of positive solutions of a non-linear heat equation, Houston J. Math.13 (1987), 39–50.MathSciNetGoogle Scholar
  2. [2]
    Escobedo, O. Kavian and H. Matano,Large time behavior of solutions of a dissipative nonlinear heat equation, Comm. Partial Differntial Equations20 (1995), 1427–1452.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    H. Fujita,On the blowing-up of solutions of the Cauchy problem for u t = δu + u1+α, J. Fac. Sci. Univ. Tokyo Sect. I13 (1996), 109–124.Google Scholar
  4. [4]
    V. A. Galaktionov, S. P. Kurdyumov and A. A. Samarskii,On asymptotic eigenfunctions of the Cauchy problem for a nonlinear parabolic equation, Math. USSR Sbomik54 (1986), 421–455.MATHCrossRefGoogle Scholar
  5. [5]
    J. Ginibre and G. Velo,The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. I. Compactness methods, Physica D95 (1996), 191–228.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    J. Ginibre and G. Velo,The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. II. Contraction methods, Comm. Math. Phys.187 (1997), 45–79.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    A. Gmira and L. Veron,Large time behavior of the solutions of a semilinear parabolic equation in R N, J. Differential Equations53 (1984), 258–276.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    K. Hayakawa,On non-existence of global solutions of some semi-linear parabolic differential equations, Proc. Japan Acad.49 (1973), 503–505.MATHMathSciNetGoogle Scholar
  9. [9]
    N. Hayashi, E. I. Kaikina and P. I. Naumkin,Large time behavior of solutions to the dissipative nonlinear Schrödinger equation, Proc. Roy. Soc. Edinburgh130A (2000), 1029–1043.CrossRefMathSciNetGoogle Scholar
  10. [10]
    N. Hayashi, E. I. Kaikina and P. I. Naumkin,Large time behavior of solutions to the Landau-Ginzburg type equation, Funkcial. Ekvac.44 (2001), 171–200.MATHMathSciNetGoogle Scholar
  11. [11]
    S. Kamin and L. A. Peletier,Large time behaviour of solutions of the heat equation with absorption, Ann. Scuola Norm. Sup. Pisa12 (1985), 393–408.MATHMathSciNetGoogle Scholar
  12. [12]
    O. Kavian,Remarks on the large time behavior of a nonlinear diffusion equation, Ann. Inst. H. Poincaré Anal. Non Linéaire4 (1987), 423–452.MATHMathSciNetGoogle Scholar
  13. [13]
    K. Kobayashi, T. Sirao and H. Tanaka,On the growing up problem for semi-linear heat equations, J. Math. Soc. Japan29 (1977), 407–424.MATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    N. Okazawa and T. Yokota,Perturbation theorems for m-accretive operators applied to the nonlinear Schrödinger and complex Ginzburg-Landau equations, J. Math. Soc. Japan54 (2002), 1–19.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    F. B. Weissler,Existence and non-existence of global solutions to a nonlinear heat equation, Israel J. Math.38 (1988), 29–40.CrossRefMathSciNetGoogle Scholar

Copyright information

© 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

Personalised recommendations