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

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1912.05922.pdf (615.0Kb)
Date
2020
Publisher city
Paris
Publisher
Cahier de recherche CEREMADE, Université Paris-Dauphine
Publishing date
01-2020
Link to item file
https://hal.archives-ouvertes.fr/hal-02447669
Dewey
Analyse
Sujet
Blow-up profile; Complex Ginzburg-Landau equation
URI
https://basepub.dauphine.fr/handle/123456789/20717
Collections
  • CEREMADE : Publications
Metadata
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Author
Duong, Giao Ky
Nouaili, Nejla
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Zaag, Hatem
581232 Laboratoire Analyse, Géométrie et Applications [LAGA]
Type
Document de travail / Working paper
Item number of pages
85
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.

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