Global propagation of analyticity and unique continuation for semilinear conservative PDEs

TL;DR

This paper reviews recent methods for establishing global unique continuation in conservative PDEs by propagating analyticity, with applications to wave, plate, and Schrödinger equations, based on finite determining modes.

Contribution

It introduces a new abstract method leveraging finite determining modes to prove global analyticity propagation and unique continuation for certain conservative PDEs.

Findings

Established global unique continuation for semilinear wave, plates, and Schrödinger equations.

Developed a new method based on propagation of analyticity.

Connected finite determining modes to unique continuation properties.

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{\'e}es EDP 2025}.