Exponential attractor for complex Ginzburg-Landau equation in three-dimensions. (English) Zbl 1041.35017
The authors deal with the long-time behaviour of the complex Ginzburg-Landau equation in three spatial dimensions, that is
\[
\begin{gathered} u_t=\rho u+ (1+ i\gamma)\Delta u- (1+ i\mu)| u|^{2\sigma}u,\\ u(0,x)= u_0(x),\end{gathered}\tag{1}
\]
where \(u\) is an unknown complex-valued function defined in the \((3+1)\)-dimensional space-time \(\mathbb{R}^{3+1}\), \(\Delta\) is a Laplacian in \(\mathbb{R}^3\), \(\rho> 0\), \(\gamma\), \(\mu\) are real parameters; \(\Omega\subset \mathbb{R}^3\) is a bounded domain. Based on the squeezing property for a semigroup for (1) the authors prove the existence of the exponential attractor.
Reviewer: Messoud A. Efendiev (Berlin)
MSC:
35B41 | Attractors |
37L30 | Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems |
35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
35G25 | Initial value problems for nonlinear higher-order PDEs |
35Q55 | NLS equations (nonlinear Schrödinger equations) |