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Semilinear Elliptic Problems Associated with the Complex Ginzburg-Landau Equation

  • Noboru Okazawa
Part of the Operator Theory: Advances and Applications book series (OT, volume 168)

Abstract

The complex Ginzburg-Landau equation is characterized by complex coefficients with positive real parts in front of both the Laplacian and the nonlinear term. Let k+iβ be the coefficient of the nonlinear term |u|q−2 u with q ≥ 2. If \( \kappa ^{ - 1} |\beta | \leqslant 2\sqrt {q - 1} /(q - 2) \), then the family of solution operators forms a semigroup of (quasi-)contractions on L 2(Ω), Ω ⊂ ℝN. Under an additional condition 2 ≤ q ≤ 2 + N/4 the family forms a semigroup of locally Lipschitz operators even if \( \kappa ^{ - 1} |\beta | > 2\sqrt {q - 1} /(q - 2) \) . As a beginning to understand such semigroups the resolvent problem for the equation is developed.

Keywords

Strong Solution Monotone Operator Maximal Monotone Solution Operator Maximal Monotone Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Mathematics Studies, vol. 5, North-Holland/American Elsevier, Amsterdam/New York, 1973.Google Scholar
  2. [2]
    H. Brézis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983.zbMATHGoogle Scholar
  3. [3]
    Ph. Clément and P. Egberts, On the sum of two maximal monotone operators, Differential Integral Equations 3 (1990), 1127–1138.zbMATHMathSciNetGoogle Scholar
  4. [4]
    P. Egberts, On the sum of accretive operators, Ph.D. thesis, TU Delft, 1992.Google Scholar
  5. [5]
    Y. Giga and M. Giga, Nonlinear Partial Differential Equations, Asymptotic Behavior of Solutions and Self-Similar Solutions, Kyoritsu Shuppan, Tokyo, 1999 (in Japanese); English Translation, Birkhäuser, Basel, 2005.Google Scholar
  6. [6]
    T. Kato, Perturbation Theory for Linear Operators, Grundlehren Math. Wissenschaften, vol. 132, Springer Verlag, Berlin and New York, 1966.Google Scholar
  7. [7]
    T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508–520.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    Y. Kobayashi and N. Tanaka, Semigroups of Lipschitz operators, Adv. Differential Equations 6 (2001), 613–640.zbMATHMathSciNetGoogle Scholar
  9. [9]
    N. Okazawa and T. Yokota, Monotonicity method applied to the complex Ginzburg-Landau and related equations, J. Math. Anal. Appl. 267 (2002), 247–263.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    N. Okazawa and T. Yokota, Global existence and smoothing effect for the complex Ginzburg-Landau equation with p-Laplacian, J. Differential Equations 182 (2002), 541–576.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    N. Okazawa and T. Yokota, Non-contraction semigroups generated by the complex Ginzburg-Landau equation, Nonlinear Partial Differential Equations and Their Applications (Shanghai, 2003), 490–504, GAKUTO Internat. Ser. Math. Sci. Appl., vol. 20, Gakkōtosho, Tokyo, 2004.Google Scholar
  12. [12]
    N. Okazawa and T. Yokota, General nonlinear semigroup approach to the complex Ginzburg-Landau equation, in preparation.Google Scholar
  13. [13]
    R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Math. Surv. Mono., vol. 49, Amer. Math. Soc., Providence, RI, 1997.Google Scholar
  14. [14]
    H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Pure and Applied Math., vol. 204, Marcel Dekker, New York, 1997.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Noboru Okazawa
    • 1
  1. 1.Department of Mathematics ScienceUniversity of TokyoTokyoJapan

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