Unique continuation and approximate controllability for a degenerate parabolic equation

TL;DR

This paper establishes unique continuation properties and approximate controllability for weakly degenerate parabolic equations in one dimension using a new Carleman estimate, advancing control theory for degenerate PDEs.

Contribution

It introduces a novel local Carleman estimate and applies it to prove unique continuation and approximate controllability for weakly degenerate parabolic equations.

Findings

Solutions vanishing on the degeneracy set are identically zero.

Approximate controllability under Dirichlet boundary conditions is achieved.

A new Carleman estimate tailored for degenerate equations is developed.

Abstract

This paper studies unique continuation for weakly degenerate parabolic equations in one space dimension. A new Carleman estimate of local type is obtained to deduce that all solutions that vanish on the degeneracy set, together with their conormal derivative, are identically equal to zero. An approximate controllability result for weakly degenerate parabolic equations under Dirichlet boundary condition is deduced.