Global propagation of analyticity and unique continuation for semilinear waves

TL;DR

This paper introduces a novel method combining control theory and Galerkin approximation to prove global analyticity propagation and unique continuation for semilinear wave equations, with implications for boundary control and observability.

Contribution

It develops a new approach to demonstrate analyticity propagation and unique continuation for semilinear wave-type equations, extending control theory techniques to nonlinear PDEs.

Findings

Analyticity can be propagated from a subset satisfying geometric control conditions.

Unique continuation holds when solutions are zero on the control subset.

Observability estimates are obtained for subcritical, defocusing nonlinearities.

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 $\omega$ 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 Schr\''odinger equation.