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Global propagation of analyticity and unique continuation for semilinear waves
[Propagation globale de l’analyticité et prolongement unique pour des ondes semi-linéaires]
Camille Laurent1  ; Cristóbal Loyola2
1CNRS UMR 9008, Université Reims-Champagne-Ardennes, Laboratoire de Mathématiques de Reims (LMR), Moulin de la Housse-BP 1039, 51687 Reims cedex 2, France
2Sorbonne Université, Université Paris Cité, CNRS, Laboratoire Jacques-Louis Lions, LJLL, F-75005 Paris, France
Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 1323-1387

In this article, we develop a new method to prove both global propagation of analyticity and unique continuation in finite time for solutions of semilinear wave-type equations with analytic nonlinearity. It combines control theory techniques and Galerkin approximation, inspired by Hale-Raugel [41], to prove that analyticity in time can be propagated for the nonlinear equation from a zone where linear observability holds towards the full space. For semilinear wave equations on a bounded domain, this implies that analyticity can be propagated to the entire domain from a subset $\omega $ that satisfies the geometric control condition. It also implies the unique continuation when the solution is assumed to be zero on $\omega $. When the nonlinearity is assumed to be subcritical and defocusing, we also obtain observability estimates in the optimal time of the geometric control condition. For semilinear plate equations, similar propagation of analyticity is achieved by assuming the controllability of the linear Schrödinger equation.

Dans cet article, nous développons une nouvelle méthode pour établir à la fois la propagation globale de l’analyticité et le prolongement unique en temps fini pour les solutions d’équations semi-linéaires de type onde à non-linéarité analytique. Combinant des techniques de théorie du contrôle et une approximation de Galerkin, dans l’esprit de Hale-Raugel [41], elle permet de propager l’analyticité en temps de l’équation non linéaire, depuis une zone où l’observabilité linéaire est vérifiée, vers l’espace tout entier. Pour les équations des ondes semi-linéaires sur un domaine borné, cela implique que l’analyticité peut être propagée au domaine tout entier à partir d’un sous-ensemble $\omega $ satisfaisant la condition de contrôle géométrique. Cela implique également le prolongement unique lorsque la solution est supposée nulle sur $\omega $. Lorsque la non-linéarité est supposée sous-critique et défocalisante, nous obtenons aussi des estimations d’observabilité dans le temps optimal de la condition de contrôle géométrique. Pour les équations des plaques semi-linéaires, un résultat analogue de propagation de l’analyticité est obtenu en supposant la contrôlabilité de l’équation de Schrödinger linéaire.

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Accepté le :
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DOI : 10.5802/jep.348
Classification : 35A20, 35B60, 93B07, 35L71, 35L75
Keywords: Propagation of analyticity, unique continuation, semilinear wave equation, semilinear plate equation
Mots-clés : Propagation de l’analyticité, prolongement unique, équation des ondes semi-linéaire, équation des plaques semi-linéaire

Camille Laurent  1   ; Cristóbal Loyola  2

1 CNRS UMR 9008, Université Reims-Champagne-Ardennes, Laboratoire de Mathématiques de Reims (LMR), Moulin de la Housse-BP 1039, 51687 Reims cedex 2, France
2 Sorbonne Université, Université Paris Cité, CNRS, Laboratoire Jacques-Louis Lions, LJLL, F-75005 Paris, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Camille Laurent; Cristóbal Loyola. Global propagation of analyticity and unique continuation for semilinear waves. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 1323-1387. doi: 10.5802/jep.348
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