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)


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.


Strong Solution Monotone Operator Maximal Monotone Solution Operator Maximal Monotone Operator 
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© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

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

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