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

TL;DR

This paper establishes Carleman estimates for degenerate parabolic equations with interior degeneracy and applies these to controllability problems, demonstrating null, approximate controllability, and unique continuation depending on control region placement.

Contribution

It introduces new Carleman estimates tailored for interior degeneracy and applies them to control theory for degenerate parabolic equations, covering different control scenarios.

Findings

Null controllability when the degeneracy point is in the control region

Unique continuation when the degeneracy point is outside the control region

Existence of solutions via Galerkin method for the studied equations

Abstract

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 $0\notin\omega_0$.