Global propagation of analyticity and unique continuation for semilinear conservative PDEs

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}.