Observability properties of the singular Grushin equation

We study the observability properties of the Grushin equation with an inverse square potential, whose singularity occurs at the boundary of two-dimensional rectangular domains or in the interior of the domain in higher dimensions. In some specific configurations of the observation set, we obtain the exact minimal time of observability. The analysis we present relies on recent Carleman estimates obtained by K. Beauchard, J. Dardé, and S. Ervedoza. As a byproduct of these results, we observe, for the heat equation associated to the Laplace-Beltrami operator on almost-Riemannian manifolds, a dependence of the minimal time of observability on the dimension of the singularity.

💡 Research Summary

The paper investigates the observability properties of heat equations driven by Grushin‑type operators perturbed with an inverse‑square potential. The domain is a product Ω=Ωₓ×Ω_y, where Ωₓ⊂ℝ^{dₓ} (dₓ=1 or dₓ≥3) and Ω_y is a compact Riemannian manifold of dimension d_y≥1. The studied equation reads

∂_t f – Δₓ f – |x|^{2γ} Δ_y f + ν² – H|x|^{–2} f = 0, (t,x,y)∈(0,T)×Ω,

with homogeneous Dirichlet boundary conditions. Here γ≥1, ν>0 are parameters, and H is the optimal constant in the Hardy inequality

∫{Ωₓ} |u|²|x|^{–2} ≤ H ∫{Ωₓ} |∇ₓ u|², u∈H₀¹(Ωₓ).

When 0 belongs to the boundary of Ωₓ, H=dₓ²/4; when 0 lies in the interior and dₓ≥3, H=(dₓ–2)²/4. The presence of the singular term ν²–H|x|^{–2} makes the operator non‑self‑adjoint in the standard L²‑space, but the Hardy inequality guarantees well‑posedness.

The main goal is to determine the minimal observation time

T(ω) = inf{ T>0 | the system is observable from an open set ω⊂Ω in time T },

where observability means the existence of C>0 such that

∥f(T)∥{L²(Ω)}² ≤ C ∫₀^{T}∫{ω} |f(t)|².

Two families of results are proved.

Theorem 1.2 (γ=1).
Assume γ=1, ν>0 and either H1 (dₓ=1, Ωₓ=(0,L)) or H2 (dₓ≥3, 0∈Ωₓ). Let ω=ωₓ×Ω_y be an open set that does not contain the singular point 0. If ωₓ is a “ring” around 0, i.e. there exist 0<r<R with ωₓ={r<|x|<R}, then

 T(ω) = r² /