Analyticity and observability for fractional heat equation on $\mathbb{R}^n$

TL;DR

This paper investigates the analytic properties and observability inequalities of solutions to fractional heat equations on the entire space, establishing bounds, unique continuation, and observability results with explicit dependence on the coefficients.

Contribution

It provides new quantitative bounds and observability inequalities for fractional heat equations with analytic coefficients, including explicit dependence on the analytic radius.

Findings

Solutions have a uniform positive analytic radius for all time.

Log-type ultra-analytic bounds are established when coefficients are ultra-analytic.

An observability inequality from a thick set is proved for the fractional heat equation.

Abstract

In this paper, we study quantitative spatial analytic bounds and unique continuation inequalities of solutions for fractional heat equations with an analytic lower order term on the whole space. At first, we show that the solution has a uniform positive analytic radius for all time, and the solution enjoys a log-type ultra-analytic bound if the coefficient is ultra-analytic. Second, we prove a H\"{o}lder type interpolation inequality on a thick set, with an explicit dependence on the analytic radius of coefficient. Finally, by the telescoping series method, we establish an observability inequality from a thick set. As a byproduct of the proof, we obtain observability inequalities in weighted spaces from a thick set for the classical heat equation with a lower order term.