In various phenomena such as pattern formation, neural firing in the brain
and cell migration, interactions that can affect distant objects globally in
space can be observed. These interactions are referred to as nonlocal
interactions and are often modeled using spatial convolution with an
appropriate integral kernel. Many evolution equations incorporating nonlocal
interactions have been proposed. In such equations, the behavior of the system
and the patterns it generates can be controlled by modifying the shape of the
integral kernel. However, the presence of nonlocality poses challenges for
mathematical analysis. To address these difficulties, we develop an
approximation method that converts nonlocal effects into spatially local
dynamics using reaction-diffusion systems. In this paper, we present an
approximation method for nonlocal interactions in evolution equations based on
a linear sum of solutions to a reaction-diffusion system in high-dimensional
Euclidean space up to three dimensions. The key aspect of this approach is
identifying a class of integral kernels that can be approximated by a linear
combination of specific Green functions in the case of high-dimensional spaces.
This enables us to demonstrate that any nonlocal interactions can be
approximated by a linear sum of auxiliary diffusive substances. Our results
establish a connection between a broad class of nonlocal interactions and
diffusive chemical reactions in dynamical systems.

Este artículo explora los viajes en el tiempo y sus implicaciones.

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