Observability from measurable sets for strongly coupled parabolic systems via single-component observation

TL;DR

This paper proves an observability inequality for strongly coupled parabolic systems from space-time measurable sets, overcoming high-frequency oscillation challenges with a new integral-type inequality.

Contribution

It introduces a novel integral-type interpolation observability inequality for coupled parabolic systems, enabling observability from measurable sets despite high-frequency oscillations.

Findings

Established observability inequality from measurable sets for coupled parabolic systems.

Developed a new integral-type interpolation inequality based on a Remez-type inequality.

Extended strategies for observability to strongly coupled systems with complex coefficients.

Abstract

We establish an observability inequality from space-time measurable sets for a class of strongly coupled parabolic systems consisting of two equations, where the observation acts on a single-component. The model is motivated by parabolic equations with complex coefficients and serves as a prototypical example of strongly coupled systems. The main difficulty lies in the fact that, unlike in the scalar and weakly coupled cases, pointwise-in-time interpolation observability estimates fail, as the observed component may exhibit high-frequency oscillatory cancellations induced by the coupling. To overcome this difficulty, we develop a new integral-type interpolation observability inequality based on a Remez-type inequality. With the aid of this integral-type interpolation observability inequality and the strategy developed in [Phung and Wang, JEMS, (2013), 681--703] and [Apraiz, Escauriaza,Wang and Zhang, JEMS, (2014), 2433--2475] for deriving observability from measurable sets, we obtain the desired observability inequality.