# Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function

U. Biccari, E. Zuazua
J. Differential Equations, Vol. 261, No. 5 (2016), pp. 2809-2853,DOI: 10.1016/j.jde.2016.05.019

Abstract: This article is devoted to the analysis of control properties for a heat equation with a singular potential $\mu/\delta^2$, defined on a bounded $C^2$ domain $\Omega\subsetR^N$, where δ is the distance to the boundary function. More precisely, we show that for any μ≤1/4 the system is exactly null controllable using a distributed control located in any open subset of Ω, while for μ>1/4 there is no way of preventing the solutions of the equation from blowing-up. The result is obtained applying a new Carleman estimate.