Characterizations of multidimensional compact almost automorphic functions and applications to Poissons and heat equations

Let $\mathcal{G}$ be a non-empty subset of the Euclidean space $\mathbb{R}^m$ ($m \geq 1$). This work is dedicated to further exploring the properties of $\mathcal{G}$-multi-almost automorphic functions defined on $\mathbb{R}^m$ with values in a Banach space $\mathbb{X}$. Using the theory of $\mathcal{G}$-multi-almost automorphic functions, we provide two new characterizations of compact almost automorphic functions. In the first characterization, $\mathcal{G}$ corresponds to the lattice subgroup $\mathbb{Z}^m \subset \mathbb{R}^m$; in the second, $\mathcal{G}$ is taken to be a dense subset of $\mathbb{R}^m$. Furthermore, we establish the invariance of the space of bounded and compactly $\mathcal{G}$-multi-almost automorphic functions under integral operators with Bi-almost automorphic kernels. Finally, we present applications to the analysis of the almost automorphic dynamics of Poisson's equation and the heat equation.