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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
##### Citations:
Zbl 0889.35045; Zbl 1005.35086
Full Text:
##### References:
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