Null controllability for a heat equation with a singular inverse-square   potential involving the distance to the boundary function
  This article is devoted to the analysis of control properties for a heat equation with singular potential $\mu/\delta^2$, defined on a bounded $C^2$ domain $\Omega\subset\mathbb{R}^N$, where $\delta$ is the distance to the boundary function. More precisely, we show that for any $\mu\leq 1/4$ the system is exactly null controllable using a distributed control located in any open subset of $\Omega$, while for $\mu>1/4$ there is no way of preventing the solutions of the equation from blowing-up. The result is obtained applying a new Carleman estimate. 
Umberto Biccari, Enrique Zuazua
math.AP