Global propagation of analyticity and unique continuation for semilinear waves

Abstract

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, 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 with Dirichlet boundary condition on a bounded domain, this implies that analyticity can be propagated to the entire domain from a subset ω that satisfies the geometric control condition. It also implies the unique continuation when the solution is assumed to be zero on ω. 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 Schr\"odinger equation.

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Cite

BibTeX RIS

BibTeX

@article{laurent2026global,
  title = {{Global propagation of analyticity and unique continuation for semilinear waves}},
  author = {Camille Laurent and Cristóbal Loyola},
  journal = {Journal de l'École polytechnique — Mathématiques},
  year = {2026},
  month = aug,
  doi = {10.5802/jep.348},
  eprint = {2407.02866},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-04632284v2},
  note = {CC BY 4.0},
}

RIS

TY  - JOUR
TI  - Global propagation of analyticity and unique continuation for semilinear waves
AU  - Camille Laurent
AU  - Cristóbal Loyola
PY  - 2026
DA  - 2026/08/27
JO  - Journal de l'École polytechnique — Mathématiques
DO  - 10.5802/jep.348
UR  - https://hal.science/hal-04632284v2
AB  - 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, 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 with Dirichlet boundary condition on a bounded domain, this implies that analyticity can be propagated to the entire domain from a subset ω that satisfies the geometric control condition. It also implies the unique continuation when the solution is assumed to be zero on ω. 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 Schr\"odinger equation.
SN  - 2429-7100
SN  - 2270-518X
ER  -

Abstract

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