Observability for a Jordan-Type Cross-Diffusion System in the Whole Space

In this article, we study the observability of a linear cross-diffusion system in [Formula: see text] with a Jordan-type diffusion matrix. The system cannot be reduced to independent heat equations. We prove that the thickness of the observation set [Formula: see text] is equivalent to both a Hölder-type interpolation inequality and an observability inequality for this system. This extends the classical thick-set framework from scalar heat equations to parabolic systems with nondiagonalizable diffusion matrices. We also provide counterexamples demonstrating that the observability inequality fails when only one component of the state is observed.