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Construction of a blow-up solution for the Complex Ginzburg-Landau equation in a critical case, β≠0

Duong, Giao Ky; Nouaili, Nejla; Zaag, Hatem (2022), Construction of a blow-up solution for the Complex Ginzburg-Landau equation in a critical case, β≠0. https://basepub.dauphine.fr/handle/123456789/20717

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1912.05922.pdf (615.0Kb)
Type
Document de travail / Working paper
External document link
https://hal.archives-ouvertes.fr/hal-02447669
Date
2022
Series title
Cahier de recherche CEREMADE, Université Paris Dauphine-PSL
Published in
Paris
Pages
85
Metadata
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Author(s)
Duong, Giao Ky
Nouaili, Nejla
Zaag, Hatem cc
Abstract (EN)
We construct a solution for the Complex Ginzburg-Landau (CGL) equation in ageneral critical case, which blows up in finite time T only at one blow-up point. We also give a sharp description of its profile. In a first part, we construct formally a blow-up solution. In a second part we give the rigorous proof. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude. The interpretation of the parameters of the finite dimension problem in terms of the blow-up point and time allows to prove the stability of the constructed solution. We would like to mention that the asymptotic profile of our solution is different from previously known profiles for CGL or for the semilinear heat equation.
Subjects / Keywords
Blow-up profile; Complex Ginzburg-Landau equation

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