Abstract
This paper is devoted to the observability of a class of two-dimensional Kolmogorov-type equations presenting a quadratic degeneracy. We give lower and upper bounds for the critical time. These bounds coincide in symmetric settings, giving a sharp result in these cases. The proof is based on Carleman estimates and on the spectral properties of a family of non-selfadjoint Schr\"odinger operators, in particular the localization of the first eigenvalue and Agmon type estimates for the corresponding eigenfunctions.
Highlights
Abstract . — This paper is devoted to the observability of a class of two-dimensional Kolmogorov-type equations presenting a quadratic degeneracy
We prove Agmon-type estimates for the first eigenfunction, which gives the negative result for T < Tmin, and we estimate the decay of the corresponding semigroup
We show that the solutions of (2.1) for n ∈ Z give the Fourier coefficients of a solution of (1.1)
Summary
We describe the strategy for the proof of Theorem 1.5. We only give the main ideas, and the details will be postponed to the following two sections. — We begin the proof of Theorem 1.5 with the first statement and prove observability for (1.1) when T > Tmax. With this result, it is enough to prove Proposition 2.5 for n large. There exist n0 ∈ N and C > 0 such that for n n0 and a solution un of (2.1) one has un(τ2) The proof of this proposition is based on carefully constructed Carleman estimates, in the spirit of [BDE20]. It is enough to prove Propositions 2.3, 2.6 and 2.7 to get the observability of (1.1) through Γ for T > Tmax |ψn(− −)|2 + |ψn( +)|2 Cn exp − 2n(1 − ε)q (0)Tmin With this proposition we prove that we cannot have observability through Γ in time T < Tmin. Spectral properties of the Kolmogorov equation we prove Propositions 2.1, 2.2, 2.3 and 2.8
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