A generalized complex Ginzburg-Landau equation: Global existence and stability results

Simão Correia; Mário Figueira

doi:10.3934/cpaa.2021056

Article Contents

2021, Volume 20, Issue 5: 2021-2038. Doi: 10.3934/cpaa.2021056

This issue Previous Article Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation Next Article Global well-posedness for effectively damped wave models with nonlinear memory

A generalized complex Ginzburg-Landau equation: Global existence and stability results

Simão Correia^1, , and
Mário Figueira^2,

1.
Center for Mathematical Analysis, Geometry and Dynamical Systems, Department of Mathematics, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
2.
CMAF-CIO, Universidade de Lisboa, Edifício C6, Campo Grande, 1749-016 Lisboa, Portugal

^* Corresponding author
^* Corresponding author

Received: October 2020

Revised: February 2021

Early access: April 2021

Published: May 2021

The first author is supported by Fundação para a Ciência e Tecnologia, through the grant UIDB/MAT/04459/2020. The second author is supported by Fundação para a Ciência e Tecnologia, through the grant UIDB/04561/2020

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Abstract

We consider the complex Ginzburg-Landau equation with two pure-power nonlinearities and a damping term. After proving a general global existence result, we focus on the existence and stability of several periodic orbits, namely the trivial equilibrium, bound-states and solutions independent of the spatial variable. In particular, we construct bound-states either explicitly in the real line or through a bifurcation argument for a double eigenvalue of the Dirichlet-Laplace operator on bounded domains.

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Mathematics Subject Classification: Primary: 35Q56, 35B10, 35B35.

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