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 (2020), 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
TypeDocument de travail / Working paper
External document linkhttps://hal.archives-ouvertes.fr/hal-02447669
Cahier de recherche CEREMADE, Université Paris-Dauphine
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Author(s)Duong, Giao Ky
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Laboratoire Analyse, Géométrie et Applications [LAGA]
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 / KeywordsBlow-up profile; Complex Ginzburg-Landau equation
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