Reaction, diffusion and nonlocal interactions in high-dimensional space

Hiroshi Ishii; Yoshitaro Tanaka

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arXiv · 2504.15180

Reaction, diffusion and nonlocal interactions in high-dimensional space

Hiroshi Ishii·Yoshitaro Tanaka

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Abstract

In this paper we consider the mathematical relationship between nonlocal interactions of convolution type and multiple diffusive substances in high dimensions. Motivated by that the nonlocal evolution equations reproduce similar patterns to those in reaction-diffusion systems, we approximate nonlocal interactions in evolution equations by the solution to a reaction-diffusion system in any dimensional Euclidean space. The key aspect of this approach is that any absolutely integrable radial kernels can be approximated by a linear combination of specific Green functions. This enables us to demonstrate that any nonlocal interactions of convolution type can be approximated by a linear sum of auxiliary diffusive substances. Moreover, we show that the parameters in the reaction-diffusion system can be specified depending on the kernel shape up to three dimensions. Our results establish a connection between a broad class of nonlocal interactions and diffusive chemical reactions in dynamical systems.

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Hiroshi Ishii, Yoshitaro Tanaka. 2025-04-21. Reaction, diffusion and nonlocal interactions in high-dimensional space. https://arxiv.org/abs/2504.15180

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