Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions

TL;DR

This paper develops Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions, establishing new observability inequalities and addressing a previously unstudied problem.

Contribution

It introduces the first Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions, advancing control theory in this context.

Findings

Established well-posedness of the problem

Derived new Carleman estimates for the adjoint problem

Proved observability inequalities for the system

Abstract

We consider a parabolic problem with degeneracy in the interior of the spatial domain and Neumann boundary conditions. In particular, we will focus on the well-posedness of the problem and on Carleman estimates for the associated adjoint problem. The novelty of the present paper is that for the first time it is considered a problem with an interior degeneracy and Neumann boundary conditions so that no previous result can be adapted to this situation. As a consequence new observability inequalities are established.