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A simple approach to the Cauchy problem for complex Ginzburg-Landau equations by compactness methods. (English) Zbl 1248.35203
Summary: This paper is concerned with the Cauchy problem (CGL) in \(L^2(\mathbb{R}^N)\) for complex Ginzburg-Landau equations with Laplacian \(\Delta\) and nonlinear term \(|u|^{q-2} u\) multiplied by the complex coefficients \(\lambda+ i\alpha\) and \(\kappa+ i\beta\), respectively (\(q\geq2\), \(\lambda> 0\), \(\kappa> 0\), \(\alpha,\beta\in\mathbb{R}\)). The global existence of strong solutions to (CGL) is established without any upper restriction on \(q\geq 2\) but with some restriction on \(\alpha/\lambda\) and \(\beta/\kappa\). The result corresponds to J. Ginibre and G. Velo [Physica D 95, No. 2–4, 191–228 (1996; Zbl 0889.35045), 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 \(\Delta\) replaced with \(\Delta- V_R\), where \(VR(x):= (|x|-R)^2\, (|x|>R)\), \(VR(x):= 0\, (|x|\leq R)\). The solvability of (CGL)\(_R\) is a direct consequence of N. Okazawa and T. Yokota [J. Differ. Equations 182, No. 2, 541–576 (2002; Zbl 1005.35086), Theorem 4.1]. Taking the limit of global strong solutions to (CGL)\(_R\) as \(R\to\infty\) yields a global strong solution to (CGL). The result gives also an unbounded version of Okazawa and Yokota [loc. cit., Theorem 1.1 with \(p=2\)] for the initial-boundary value problem on bounded domains.

MSC:
35Q56 Ginzburg-Landau equations
47H20 Semigroups of nonlinear operators
35D35 Strong solutions to PDEs
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