A simple approach to the Cauchy problem for complex Ginzburg-Landau equations by compactness methods

作者：Clement Philippe; Okazawa Noboru; Sobajima Motohiro; Yokota Tomomi^*

来源：Journal of Differential Equations, 2012, 253(4): 1250-1263.

DOI：10.1016/j.jde.2012.05.002

• Compactness methods
• Cauchy problems
• Complex Ginzburg-Landau equations
• Subdifferential operators

This paper is concerned with the Cauchy problem (CGL) in L-2(R-N) for complex Ginzburg-Landau equations with Laplacian Delta and nonlinear term vertical bar u vertical bar(q-2)u multiplied by the complex coefficients lambda + i alpha and kappa + i beta respectively (q %26gt;= 2, lambda %26gt; 0, kappa %26gt; 0, alpha, beta is an element of R). The global existence of strong solutions to (CGL) is established without any upper restriction on q %26gt;= 2 but with some restriction on alpha/lambda and beta/kappa. The result corresponds to Ginibre and Velo (1996) [3, Proposition 5.1] which is technically proved by combining convolution (regularizing) methods with compactness (localizing) methods, while our proof here is fairly simplified. The key to our proof is the Cauchy problem (CGL)R which is (CGL) with A replaced with Delta - V-R, where V-R(X) := (vertical bar x vertical bar - R)(2) (vertical bar x vertical bar %26gt; R), V-R(x) := 0 (vertical bar x vertical bar %26lt;= R). The solvability of (CGL)(R) is a direct consequence of Okazawa and Yokota (2002) [16, Theorem 4.1]. Taking the limit of global strong solutions to (CGL)(R) as R -%26gt; infinity yields a global strong solution to (CGL). The result gives also an unbounded version of Okazawa and Yokota (2002) [16, Theorem 1.1 with p = 2] for the initial-boundary value problem on bounded domains.

• 出版日期2012-8-15