On the unique continuation of solutions to non-local non-linear dispersive equations

We prove unique continuation properties of solutions to a large class of nonlinear, non-local dispersive equations. The goal is to show that if $u_1,u_2$ are two suitable solutions of the equation defined in $\mathbb R^n\times[0,T]$suchthatforsomenon-emptyopenset$\Omega\subset \mathbb R^n\times[0,T]$,$u_1(x,t)=u_2(x,t)$for$(x,t) \in \Omega$,then$u_1(x,t)=u_2(x,t)$forany$(x,t)\in\mathbb R^n\times[0,T]$. The proof is based on static arguments. More precisely, the main ingredient in the proofs will be the unique continuation properties for fractional powers of the Laplacian established by Ghosh, Salo and Ulhmann in \cite{GhSaUh}, and some extensions obtained here.