Critical time of observability of the multi-dimensional Baouendi-Grushin equation.

In this presentation, we will examine the observability properties of the multi-dimensional Baouendi–Grushin equation.

From an observability perspective, this degenerate parabolic equation exhibits both interesting and somewhat unexpected behavior: unlike standard parabolic heat-like equations—where observability holds for arbitrarily small times and without restrictions on the observation set—the Baouendi–Grushin system requires geometric conditions on the observation set and is only observable for sufficiently large times.

Nowadays, the two-dimensional case is the best understood, thanks to extensive research—though the exact minimal observability time remains an open question in certain geometric configurations.

In this presention, we focus on the "higher-than-two-dimensional" setting and determine the precise critical observability time in some multi-dimensional tensorized configurations. This result is achieved by combining refined observability inequalities derived from Carleman estimates, and a Lebeau–Robbiano localization strategy.

This is a joint work with Mathilda Trabut.