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

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.