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 ${\mathbb{R}}^m$ ($m\ge 1$). 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.