Global existence and asymptotics for quasi-linear one-dimensional Klein-Gordon equations with mildly decaying Cauchy data

Stingo, Annalaura

doi:10.24033/bsmf.2755

Global existence and asymptotics for quasi-linear one-dimensional Klein-Gordon equations with mildly decaying Cauchy data
[Existence globale et comportement asymptotique de petites solutions pour des équation de Klein-Gordon critiques 1D]

Stingo, Annalaura ¹

¹ Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), 99, Avenue J.-B. Clément, F-93430 Villetaneuse

Bulletin de la Société Mathématique de France, Tome 146 (2018) no. 1, pp. 155-213

Abstract (VO)
Résumé

Let $u$ be a solution to a quasi-linear Klein-Gordon equation in one-space dimension, $□ u + u = P (u, \partial_{t} u, \partial_{x} u; \partial_{t} \partial_{x} u, \partial_{x}^{2} u)$ , where $P$ is a homogeneous polynomial of degree three, and with smooth Cauchy data of size $ε \to 0$ . It is known that, under a suitable condition on the nonlinearity, the solution is global-in-time for compactly supported Cauchy data. We prove in this paper that the result holds even when data are not compactly supported but just decaying as ${〈 x 〉}^{- 1}$ at infinity, combining the method of Klainerman vector fields with a semiclassical normal forms method introduced by Delort. Moreover, we get a one term asymptotic expansion for $u$ when $t \to + \infty$ .

Soit $u$ une solution d’une équation de Klein-Gordon quasi-linéaire en dim. 1 d’espace, $□ u + u = P (u, \partial_{t} u, \partial_{x} u; \partial_{t} \partial_{x} u, \partial_{x}^{2} u)$ , où $P$ est un polynôme homogène de degré trois, avec données initiales régulières de taille $ε \to 0$ . Il est connu que, sous certaines conditions sur la non-linéarité, la solution est globale en temps pour des données initiales à support compact. Nous montrons que ce résultat est aussi vrai quand les données ne sont pas à support compact mais seulement décroissantes à l’infini comme ${〈 x 〉}^{- 1}$ , en combinant la méthode des champs de vecteurs de Klainerman avec une méthode de formes normales semi-classiques introduite par Delort. De plus, nous obtenons un développement asymptotique à un terme pour $u$ lorsque $t \to + \infty$ , prouvant ainsi un résultat de scattering modifié.

Reçu le : 2015-07-15
Révisé le : 2016-09-14
Accepté le : 2016-11-29
Publié le : 2018-07-21

MR Zbl

DOI : 10.24033/bsmf.2755

Keywords: Global solution of quasi-linear Klein-Gordon equations, Klainerman vector fields, Semiclassical Analysis.
Mots-clés : Solution globale pour des équations de Klein-Gordon quasi-linéaires, champs de vecteurs de Klainerman, analyse semiclassique

Affiliations des auteurs :

Stingo, Annalaura ¹

¹ Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), 99, Avenue J.-B. Clément, F-93430 Villetaneuse

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     author = {Stingo, Annalaura},
     title = {Global existence and asymptotics for quasi-linear one-dimensional {Klein-Gordon} equations with mildly decaying {Cauchy} data},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {155--213},
     year = {2018},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {146},
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Stingo, Annalaura. Global existence and asymptotics for quasi-linear one-dimensional Klein-Gordon equations with mildly decaying Cauchy data. Bulletin de la Société Mathématique de France, Tome 146 (2018) no. 1, pp. 155-213. doi: 10.24033/bsmf.2755

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