The null controllability 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 derivative(t)u(x, t) - div(p(x)del u(x, t)) - (mu/vertical bar x vertical bar(2))u(x, t) = f(x, t). Here mu is a real constant. It was proved in the paper of Goldstein and Zhang (8) that the equation is well-posedness when 0 <= mu <= p(1)(n - 2)(2)/4, and in this paper, we mainly consider the case 0 <= mu < (p(1)(2) /p(2))(n - 2)(2)/4, where p(1), p(2) are two positive constants which satisfy: 0 < p(1) <= p(x) <= p(2), for all x is an element of <(Omega)over bar>. We extend the specific Carleman estimates in the paper of Ervedoza [Control and stabilization properties for a singular heat equation with an inverse-square potential. Commun Partial Differ Equ. 2008;33:1996-2019] and Vancostenoble [Lipschitz stability in inverse source problems for singular parabolic equations. Commun Partial Differ Equ. 2011;36(8):1287-1317] to the system. 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/(vertical bar x vertical bar(2) + epsilon(2)), and prove that we cannot stabilize the corresponding systems uniformly with respect to epsilon > 0..