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arXiv · 2412.07644

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

Alan Chávez·Jolbyn Castañeda·Alexis R. Carranza·Kamal Khalil
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Abstract

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.

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Alan Chávez, Jolbyn Castañeda, Alexis R. Carranza, Kamal Khalil. 2024-12-10. Characterizations of multidimensional compact almost automorphic functions and applications to Poissons and heat equations. https://arxiv.org/abs/2412.07644

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