Observability and Control Property for a Singular Heat Equation with Variable Coefficients

The goal of this paper is to analyze control properties of the parabolic equation with variable coefficients in the principal part and with a singular inverse-square potential:\,$\partial_tu(x,t)-{\rm div}(p(x)\nabla u(x,t))-({\mu}/{|x|^2})u(x,t)=f(x,t).$ Here $\mu$ is a real constant . It was proved in the paper of Goldstein and Zhang (2003) that the equation is well-posedness when $0\leq{\mu\leq p_1(n-2)^2/4}$, and in this paper, we mainly consider the case $0\leq\mu<({ p_1^2}/{ p_2})(n-2)^2/4$ , where $p_1,p_2$ are two positive constants which satisfy:\, $0< p_1\leq p(x)\leq p_2 , \forall x\in\overline{\Omega}$. We extend the specific Carleman estimates in the paper of Ervedoza (2008) and Vancostenoble (2011) to the equation we consider and apply it to deduce an observability inequality for the system. By this inequality and the classical HUM method, we obtain that we can control the equation from any non-empty open subset as for the heat equation. Moreover, we will study the case $\mu>p_2(n-2)^2/4$. We consider a sequence of regularized potentials $\mu/(|x|^2+\epsilon^2),$ and prove that we cannot stabilize the corresponding systems uniformly with respect to $\epsilon>0,$ due to the presence of explosive modes which concentrate around the singularity.