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

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.