Control and Stabilization Properties for a Singular Heat Equation with an Inverse-Square Potential

Abstract

The goal of this article is to analyze control properties of parabolic equations with a singular potential − μ/|x|2, where μ is a real number. When μ ≤ (N − 2)2/4, it was proved in [19 Vancostenoble , J. , Zuazua , E. ( 2008 ). Null controllability for the heat equation with singular inverse-square potentials . J. Funct. Anal. 254 : 1864 – 1902 .[Crossref], [Web of Science ®] , [Google Scholar]] that the equation can be controlled to zero with a distributed control which surrounds the singularity. In the present work, using Carleman estimates, we will prove that this assumption is not necessary, and that we can control the equation from any open subset as for the heat equation. Then we will study the case μ > (N − 2)2/4, and prove that the situation changes completely: indeed, we will consider a sequence of regularized potentials μ/(|x|2 + ϵ2), and prove that we cannot stabilize the corresponding systems uniformly with respect to ϵ > 0, due to the presence of explosive modes which concentrate around the singularity.