Null Controllability for a Multi-Dimensional Degenerate Parabolic Equation with Degenerated Interior Point

In this study, we study the null controllability of a multi-dimensional degenerate parabolic equation characterized by a degenerate interior point. The control domain, which is an arbitrary inner region, does not encompass the degenerate point. To tackle this problem, we adopt a new approximation methodology. Specifically, we approximate the degenerate partial differential equations (PDEs) with a series of uniformly elliptic PDEs, notwithstanding their limited regularity. We then derive the Carleman estimate for these approximate uniformly parabolic equations and establish the observability inequality, which ultimately paves the way for demonstrating the null controllability of the system.