Carleman estimate and applications for a heat equation with multiple singularities

The goal of this article is the analyses of inverse source problem and null controllability for a heat equation with multipolar inverse-square potentials: ∑j=1mμj|x−xj|2, where x1,x2,…,xm are different points, defined in a bounded C2 domain Ω⊂Rn, and μ1,…,μm are real constants. More precisely, we show that for any μj≤(n−2)24,j=1,…,m, we can obtain a Lipschitz stability result in inverse source problems and we will show that the system is null controllable. Our proofs are mainly based on Carleman estimates, and due to multiple singularities, we give a new Carleman estimate by choosing an appropriate weight function.