October  1999, 5(4): 825-848. doi: 10.3934/dcds.1999.5.825

Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties

1. 

Fachbereich Mathematik, Universität Rostock, Universitätsplatz 1, D-18055 Rostock, F. R., Germany

2. 

Fachbereich Mathematik, Universität Rostock, Universitätsplatz 1, D-18055 Rostock

Received  January 1999 Revised  June 1999 Published  July 1999

The complex Ginzburg-Landau equation (CGL, for short)

$ \partial_t u = (1 + i\nu)\Delta u + Ru- (1 + i\mu) |u|^2 u; \quad 0\le t < \infty, x\in\Omega $,

is investigated in a bounded domain ­$\Omega\subset \mathbb R^n$ with suffciently smooth boundary. Standard boundary conditions are considered: Dirichlet, Neumann or periodic. Existence and uniqueness of global smooth solutions is established for all real parameter values $\mu$ and $\nu$ if $n\le 2$, and for certain parameter values $\mu$ and $\nu$ if $n\ge 3$. Furthermore, dynamical properties of the CGL equation, such as existence of determining nodes, are shown. The proof of existence of smooth solutions hinges on the following inequality using the $L^2(\Omega)$-duality,

$|\mathfrak Im$ $<\Delta u ,\ |u|^{p-2}u>\le (|p-2|)/(2\sqrt{p-1})\mathfrak Re$ $< -\Delta u ,\ |u|^{p-2}u >.$

Citation: Sen-Zhong Huang, Peter Takáč. Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 825-848. doi: 10.3934/dcds.1999.5.825
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