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<p>In this paper, we study the mathematical relationship between nonlocal interactions of convolution type and systems of multiple diffusive substances in high-dimensional spaces. Motivated by the observation that nonlocal evolution equations can reproduce similar patterns to those arising in reaction-diffusion systems, we approximate nonlocal interactions in evolution equations by solutions to appropriate reaction-diffusion systems with multiple components in Euclidean space of arbitrary dimension. The key idea of this approach is that any absolutely integrable radial kernel can be approximated by a linear combination of specific Green functions to elliptic partial differential equations. This enables us to demonstrate that a linear sum of auxiliary diffusive substances can approximate a broad class of nonlocal interactions of convolution type. Furthermore, for spatial dimensions up to three, we show that the parameters in the reaction-diffusion system can be explicitly determined depending on the kernel shape. Our results establish a connection between a broad class of nonlocal interactions and diffusive chemical reactions in dynamical systems.</p>

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Reaction, diffusion, and nonlocal interaction in high-dimensional space

  • 期刊论文

  • 发表日期：

    2026年4月9日

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  • Hiroshi Ishii,
  • Yoshitaro Tanaka

摘要

In this paper, we study the mathematical relationship between nonlocal interactions of convolution type and systems of multiple diffusive substances in high-dimensional spaces. Motivated by the observation that nonlocal evolution equations can reproduce similar patterns to those arising in reaction-diffusion systems, we approximate nonlocal interactions in evolution equations by solutions to appropriate reaction-diffusion systems with multiple components in Euclidean space of arbitrary dimension. The key idea of this approach is that any absolutely integrable radial kernel can be approximated by a linear combination of specific Green functions to elliptic partial differential equations. This enables us to demonstrate that a linear sum of auxiliary diffusive substances can approximate a broad class of nonlocal interactions of convolution type. Furthermore, for spatial dimensions up to three, we show that the parameters in the reaction-diffusion system can be explicitly determined depending on the kernel shape. Our results establish a connection between a broad class of nonlocal interactions and diffusive chemical reactions in dynamical systems.