Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case

Feng  Zhou; Chunyou Sun

doi:10.3934/dcdsb.2016120

Article Contents

2016, Volume 21, Issue 10: 3767-3792. Doi: 10.3934/dcdsb.2016120

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Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case

Feng Zhou^1, and
Chunyou Sun^1,

1.
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000

Received: December 31, 2015

Revised: February 29, 2016

Published: November 2016

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Abstract

The purpose of this article is to analyze the dynamics of the following complex Ginzburg-Landau equation \begin{align*} \partial_{t}u-(\lambda+i\alpha)\Delta u+(\kappa+i\beta)|u|^{p-2}u-\gamma u=f(t) \end{align*} on non-cylindrical domains, which are obtained by diffeomorphic transformation of a bounded base domain, without any upper restriction on $p>2$, only with some restriction on $\beta/\kappa$. We establish the existence and uniqueness of strong and weak solutions as well as some energy inequalities for this equation on variable domains. Moreover the existence of a $\mathscr{D}$-pullback attractor is established for the process generated by the weak solutions under a slightly weaker condition that the measure of the spatial domains in the past is uniformly bounded above.

Keywords:

Complex Ginzburg-Landau equation,
non-cylindrical domains,
non-autonomous,
$\mathscr{D}$-pullback asymptotically compact,
$\mathscr{D}$-pullback attractor.

Mathematics Subject Classification: Primary: 37L30, 37N20; Secondary: 35Q56.

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