Carleman estimates for degenerate parabolic equations with single interior point degeneracy and its applications

We study the controllability of a class of $N$-dimensional degenerate parabolic equations with single interior point degeneracy. We employ the Galerkin method to prove the existence of solutions for the equations. The analysis is then divided into two cases based on whether the degenerate point $x=0$ lies within the control region ${\omega }_0$ or not. For each case, we establish specific Carleman estimates. As a result, we achieve null controllability in the first case $0\in {\omega }_0$ and unique continuation and approximate controllability in the second case $0\mathrm{\notin }{\omega }_0$.