A Fast Observability for Diffusion Equations in ${\mathbb{R}}^N$

Given an equidistributed set in the whole Euclidean space, we have established in [1] that there exists a constant positive $C$ such that the observability inequality of diffusion equations holds for all $T\in ]0,1[$, with an observability cost being of the form $C{e}^{C\mathrm{/}T}$. In this paper, for any small constant $\epsilon >0$, we prove that there exists a nontrivial equidistributed set (in the sense that whose complementary set is unbounded), so that the above observability cost can be improved to a fast form of $C{e}^{\epsilon \mathrm{/}T}$ for certain constant $C>0$. The proof is based on the strategy used in [1], as well as an interpolation inequality for gradients of solutions to elliptic equations obtained recently in [2].