Quantitative Weak Unique Continuation on Annular Domains for Backward Degenerate Parabolic Equations with Degenerate Interior Points

TL;DR

This paper proves a quantitative weak unique continuation property for backward degenerate parabolic equations with interior degeneracy, using approximation and Carleman estimates.

Contribution

It introduces a novel approach combining approximation and Carleman estimates to establish weak unique continuation for degenerate parabolic equations.

Findings

Established a quantitative weak unique continuation theorem.

Developed Carleman estimates for non-degenerate parabolic equations.

Deduced unique continuation for degenerate parabolic equations.

Abstract

In this paper, we establish a quantitative weak unique continuation theorem on an annular domain for a backward degenerate parabolic equation with a degenerate interior point. Our methodology hinges on approximating the solution of the degenerate parabolic equation through solutions of non-degenerate parabolic counterparts. Subsequently, we establish Carleman estimates for the non-degenerate parabolic equation across two separate domains. By virtue of these estimates, we deduce a quantitative weak unique continuation property for the degenerate parabolic equation, thereby substantiating the weak unique continuation result for the original degenerate parabolic equation.