Strong unique continuation for variable coefficient parabolic operators with Hardy type potential

TL;DR

This paper establishes strong unique continuation at the origin for solutions of a critical parabolic inequality with variable coefficients and Hardy-type potential, extending previous subcritical and heat operator results.

Contribution

It proves the strong unique continuation property for variable coefficient parabolic operators with Hardy potential, sharpening and extending prior results to the critical case.

Findings

Proves strong unique continuation at the origin for the inequality.

Extends previous subcritical case results to the critical case.

Improves understanding of parabolic operators with Hardy potential.

Abstract

In this paper, we prove the strong unique continuation property at the origin for solutions of the following scaling critical parabolic differential inequality \[ |\operatorname{div} (A(x,t) \nabla u) - u_t| \leq \frac{M}{|x|^{2}} |u|,\ \ \ \ \] where the coefficient matrix $A$ is Lipschitz continuous in $x$ and $t$. Our main result sharpens a previous one of Vessella concerned with the subcritical case as well as extends a recent result of one of us with Garofalo and Manna for the heat operator.