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C.D. Levermore and M. Oliver,
Distribution-valued initial data for the complex
Ginzburg-Landau equation,
Comm. Partial Differential Equations 22 (1997), 39-48.
Abstract:
The generalized complex Ginzburg-Landau (CGL) equation with a
nonlinearity of order 2s+ 1 in d spatial dimensions has
a unique local classical solution for distributional initial data in
the Sobolev space Hq provided that
q>d/2-1/s. This result directly corresponds to a
theorem for the nonlinear Schrödinger (NLS) equation which has
been proved by Cazenave and Weissler in 1990. While the proof in the
NLS case relies on Besov space techniques, it is shown here that for
the CGL equation, the smoothing properties of the linear semigroup can
be used to obtain an almost optimal result by elementary means.
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