On quantitative uniqueness for parabolic equations

TL;DR

This paper investigates quantitative uniqueness for parabolic equations with variable coefficients, establishing strong unique continuation and observability estimates under certain integrability conditions on the coefficients.

Contribution

It introduces new conditions on coefficients in Lebesgue spaces that ensure strong unique continuation and observability for parabolic equations.

Findings

Proved strong unique continuation property for parabolic equations with coefficients in specific Lebesgue spaces.

Established pointwise in time observability estimates under these conditions.

Extended the understanding of quantitative uniqueness in the context of variable coefficient parabolic PDEs.

Abstract

We consider the quantitative uniqueness properties for a parabolic type equation $ u_t-\Delta u = w(x,t) \nabla u + v(x,t) u$, when $v \in L^{p_2}_{t} L^{p_1}_x$ and $w \in L^{q_2}_{t} L^{q_1}_x$, with a suitable range for exponents $p_1$, $p_2$, $q_1$, and $q_2$. We prove a strong unique continuation property and provide a pointwise in time observability estimate.