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Smoothing effect and strong $$L^2$$-wellposedness in the complex Ginzburg-Landau equation. (English) Zbl 1110.35030
Favini, Angelo (ed.) et al., Differential equations. Inverse and direct problems. Papers of the meeting, Cortona, Italy, June 21–25, 2004. Boca Raton, FL: CRC Press (ISBN 1-58488-604-8/hbk). Lecture Notes in Pure and Applied Mathematics 251, 265-288 (2006).
In this paper it is studied the evolution problem associated to the Ginzburg-Landau equation with Dirichlet boundary condition and nonlinear term which behaves like $$| u| ^{q-2}u$$, where $$q\geq 2$$. The author establishes various results related to the $$L^2$$-strong solvability and the smoothing effect. The proofs combine monotonicity methods, elliptic and parabolic estimates, and compactness arguments.
For the entire collection see [Zbl 1098.34002].

##### MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B65 Smoothness and regularity of solutions to PDEs 35K55 Nonlinear parabolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000)