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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.

35Q55 NLS equations (nonlinear Schrödinger equations)
82D55 Statistical mechanical studies of superconductors
Full Text: DOI
[1] DOI: 10.1103/RevModPhys.65.851 · Zbl 1371.37001 · doi:10.1103/RevModPhys.65.851
[2] DOI: 10.1016/0167-2789(90)90156-J · Zbl 0702.76061 · doi:10.1016/0167-2789(90)90156-J
[3] DOI: 10.1016/0167-2789(95)00275-8 · Zbl 0885.35012 · doi:10.1016/0167-2789(95)00275-8
[4] DOI: 10.1088/0951-7715/7/4/006 · Zbl 0803.35066 · doi:10.1088/0951-7715/7/4/006
[5] DOI: 10.1016/0167-2789(94)90150-3 · Zbl 0810.35119 · doi:10.1016/0167-2789(94)90150-3
[6] DOI: 10.1016/0167-2789(87)90020-0 · Zbl 0623.58049 · doi:10.1016/0167-2789(87)90020-0
[7] DOI: 10.1016/0167-2789(96)00055-3 · Zbl 0889.35045 · doi:10.1016/0167-2789(96)00055-3
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[9] DOI: 10.1088/0951-7715/8/5/006 · Zbl 0833.35016 · doi:10.1088/0951-7715/8/5/006
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