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Global propagation of analyticity and unique continuation for semilinear conservative PDEs

Camille Laurent, Cristóbal Loyola

TL;DR

This work addresses global unique continuation and propagation of analyticity for semilinear conservative PDEs by developing an abstract observability-based framework that uses finite determining modes. The authors prove that, under the Geometric Control Condition, solutions to semilinear wave equations with analytic nonlinearities exhibit time-analyticity on the full domain once analytic behavior is observed on a subdomain, enabling global continuation and, in defocusing cases, triviality. The methodology extends to nonlinear plate and Schrödinger equations by adapting the reconstruction procedure and leveraging linear observability results and propagation of analyticity, yielding semiglobal stabilization and controllability results under natural geometric assumptions. The results advance rigidity and scattering-type questions for nonlinear dispersive and hyperbolic systems and provide a versatile tool for future applications to broader conservative PDEs.

Abstract

We review some recent results in which we develop a new method for proving global unique continuation for some conservative PDEs. The main tool is to prove some global propagation of analyticity. We first present some known results on the subject. Then, we sketch the abstract method we use, which relies on the property of finite determining modes. We give applications to semilinear wave, plates and Schr\''odinger equations. This note was written for the \emph{Proceedings of the Journ{é}es EDP 2025}.

Global propagation of analyticity and unique continuation for semilinear conservative PDEs

TL;DR

This work addresses global unique continuation and propagation of analyticity for semilinear conservative PDEs by developing an abstract observability-based framework that uses finite determining modes. The authors prove that, under the Geometric Control Condition, solutions to semilinear wave equations with analytic nonlinearities exhibit time-analyticity on the full domain once analytic behavior is observed on a subdomain, enabling global continuation and, in defocusing cases, triviality. The methodology extends to nonlinear plate and Schrödinger equations by adapting the reconstruction procedure and leveraging linear observability results and propagation of analyticity, yielding semiglobal stabilization and controllability results under natural geometric assumptions. The results advance rigidity and scattering-type questions for nonlinear dispersive and hyperbolic systems and provide a versatile tool for future applications to broader conservative PDEs.

Abstract

We review some recent results in which we develop a new method for proving global unique continuation for some conservative PDEs. The main tool is to prove some global propagation of analyticity. We first present some known results on the subject. Then, we sketch the abstract method we use, which relies on the property of finite determining modes. We give applications to semilinear wave, plates and Schr\''odinger equations. This note was written for the \emph{Proceedings of the Journ{é}es EDP 2025}.
Paper Structure (23 sections, 13 theorems, 34 equations, 2 figures)

This paper contains 23 sections, 13 theorems, 34 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Propagation of information
  3. Propagation of information for semilinear wave equation
  4. Known results of unique continuation
  5. Classical local unique continuation theorems for linear equations
  6. Some previous global results for semilinear waves
  7. Nonlinear solutions are also linear solutions
  8. Using Hörmander theorem and classical Carleman estimates
  9. Using unique continuation with partial analyticity
  10. Observability under Geometric Control Condition
  11. Main results for semilinear waves
  12. Unique continuation for semilinear waves
  13. Propagation of analyticity
  14. (ultra)Short bibliography on propagation of regularity
  15. Idea of proof
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $V$ be bounded and analytic in $t$. Let $\omega\subset \Omega$ open and $T>2\sup_{x\in\Omega}\text{dist}(x,\omega )$. Let $(u_0, u_1)\in H_0^1\times L^2(\Omega)$ and $u$ solution of that satisfies $u=0$ on $[0,T]\times \omega$. Then $u=0$ on $[0,T]\times \Omega$.

Figures (2)

  • Figure 1: The main local unique continuation theorems
  • Figure 2: The Geometric Control Condition of Bardos-Lebeau-Rauch.

Theorems & Definitions (13)

  • Theorem 2.1: Tataru-Robbiano-Zuily-Hörmander
  • Theorem 2.2: Bardos-Lebeau-Rauch BLR92
  • Theorem 3.1: Unique continuation, Laurent-Loyola LL:24
  • Theorem 3.2: Propagation of analyticity, Laurent-Loyola LL:24
  • Corollary 3.3
  • Theorem 4.1: Laurent-Loyola LL:24
  • Theorem 4.2: Nonlinear reconstruction, Laurent-Loyola LL:24
  • Lemma 4.3: A Duhamel formula/observation problem
  • Lemma 4.4
  • Theorem 5.1: Loyola Loy25:NLP
  • ...and 3 more