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The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. I: Compactness methods. (English) Zbl 0889.35045
It is studied the Cauchy problem for the equation \[ \partial_t u = \gamma u +(a+ i \alpha) \Delta u - (b + i \beta u g(|u|^2), \] \(a>0\), \(b>0\), \(g \geq 0\), in arbitrary spatial dimension. The initial data and the solutions under consideration belong to local spaces, without any restriction at infinity. It is proved the existence of solution globally defined in time with local regularity corresponding to \(L^r\), \(r \geq 2\), and \(H^1\). Some uniqueness results are presented as well.

MSC:
35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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