On the critical time of observability of the multi-dimensional Baouendi-Grushin equation

Abstract

We investigate the observability properties of the Baouendi-Grushin equation on a tensorized domain $\Omega :={\mathcal{B}}_R\times \tilde{\Omega }$, where ${\mathcal{B}}_R$ is the open ball of radius $R$ in dimension $d\ge 2$, and $\tilde{\Omega }$ is a smooth, bounded, open set of arbitrary dimension. Our main result is a precise calculation of the minimal observability time $T^{\ast }$, for tensorized observation sets of the form $\omega \times \tilde{\Omega }$, with $\omega \subset {\mathcal{B}}_R$ (internal observation), and $\Gamma \times \tilde{\Omega }$, with $\Gamma \subset \partial {\mathcal{B}}_R$ (boundary observation). The main novelty regards the sufficient condition, that is observability of the system when $T>T^{\ast }$. This is established by combining refined observability inequalities on the annulus--or the entire boundary--using Carleman estimates, together with a Lebeau-Robbiano strategy to localize the observation sets.