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Localized estimates and Cauchy problem for the logarithmic complex Ginzburg-Landau equation. (English) Zbl 0877.35116
Summary: We prove that the localization of the \(L^2\) and similar estimates for the complex Ginzburg-Landau equation \[ \partial_tu=\gamma u+(a+ i\alpha)\Delta u-(b+i\beta)ug(|u|^2), \] known to hold in the power case \(g(\rho)\sim\rho^\sigma\), also holds in the logarithmic case \(g(\rho)\sim(\text{Log }\rho)^\nu\) for large \(\rho\) with \(\nu>2\). As a consequence, the theory of the Cauchy problem in local spaces given in a previous paper also extends to the logarithmic case.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
82D55 Statistical mechanical studies of superconductors
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