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Reaction, diffusion and nonlocal interactions in high-dimensional space

Hiroshi Ishii, Yoshitaro Tanaka

TL;DR

This work demonstrates that nonlocal evolution equations with radial kernels $K$ can be faithfully approximated by multi-component reaction-diffusion systems in any dimension. By expressing $K$ as a linear combination of Green functions $k_j$ of auxiliary diffusion operators and using a singular limit on auxiliary variables, the authors establish a rigorous link between nonlocal interactions and diffusive chemical reactions, with explicit parameterization available for dimensions up to three. They provide quantitative error bounds tying the RD surrogate to the original nonlocal model and show how to approximate arbitrary radial kernels in $L^1(\mathbb{R}^n)$ via Wiener theory and polynomial approximations (Bernstein, Lagrange). The results enable practical computation and analysis of high-dimensional nonlocal dynamics through tractable RD systems, offering a unifying framework for interpreting nonlocality as quasi-steady-state diffusion among multiple species. This approach has implications for neural fields, population dynamics, and pattern formation where long-range interactions govern the observed spatiotemporal patterns.

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.

Reaction, diffusion and nonlocal interactions in high-dimensional space

TL;DR

This work demonstrates that nonlocal evolution equations with radial kernels can be faithfully approximated by multi-component reaction-diffusion systems in any dimension. By expressing as a linear combination of Green functions of auxiliary diffusion operators and using a singular limit on auxiliary variables, the authors establish a rigorous link between nonlocal interactions and diffusive chemical reactions, with explicit parameterization available for dimensions up to three. They provide quantitative error bounds tying the RD surrogate to the original nonlocal model and show how to approximate arbitrary radial kernels in via Wiener theory and polynomial approximations (Bernstein, Lagrange). The results enable practical computation and analysis of high-dimensional nonlocal dynamics through tractable RD systems, offering a unifying framework for interpreting nonlocality as quasi-steady-state diffusion among multiple species. This approach has implications for neural fields, population dynamics, and pattern formation where long-range interactions govern the observed spatiotemporal patterns.

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.

Paper Structure

This paper contains 18 sections, 28 theorems, 145 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Mathematical setting and main results
  3. Error estimates
  4. Properties of the Green function
  5. Error estimates for reaction-diffusion approximation
  6. Approximation of a kernel by the Green function in $L^1({\mathbb{R}}^n)$
  7. Kernel approximation in general dimension
  8. Error estimate for integral kernels
  9. One-dimensional case
  10. Two-dimensional case
  11. Three-dimensional case
  12. Bernstein Polynomial
  13. Lagrange Polynomial
  14. Polynomial approximation and numerical examples
  15. Reaction-diffusion approximation of nonlocal interactions
  16. ...and 3 more sections

Key Result

Theorem 1

For any $T>0$ and $1\le p\le +\infty$, there exists a unique mild solution $u\in C([0,T];BC({\mathbb{R}}^n)\cap L^p({\mathbb{R}}^n))$ to pro:non with an initial datum $u_0\in BC({\mathbb{R}}^n)\cap L^{p}({\mathbb{R}}^n)$. This mild solution $u$ belongs to $C^{1,2}((0,T]\times {\mathbb{R}}^n)$, that for any $p\le q\le +\infty$ and $t \in [0,T]$.

Figures (4)

  • Figure 1: Profile of a compact Mexican-hat function and pattern of a solution to \ref{['eq:non-growth']}. $K(x) = \mu ( a_1/(\pi R_1^3) (R_1 - |x|)\chi_{B(R_1)}(x) - a_2/(\pi R_2^3) (R_2 - |x|)\chi_{B(R_2)}(x) )$ where $0<R_1<R_2$ is radius, $B(R)$ is a ball with radius $R$, $\chi_{B(R)}$ is the characteristic function, and $a_1, a_2$ and $\mu$ are constant. (a) $R_1=1, R_2=1.5,a_1=a_2=\mu=1$. (b) A solution of \ref{['eq:non-growth']} with $R_1=1.5, R_2=3, a_1=1.2, a_2=1, \mu=50$, and $f(u)=2u(1-u^2)$.
  • Figure 2: Graphs of $K(x)=e^{-|x|^2}$ and $K_{N+1}$ with $N=30$ (top panels), the distributions of $\{\alpha_j\}_{1\le j\le N+1}$ (bottom panels) by using the Bernstein polynomial. The graph of $K$ and $K_{N+1}$ are shown by the solid line and the dashed line, respectively. The numerical examples correspond to the results with $n=1$ (left), $n=2$ (middle), and $n=3$ (right), respectively.
  • Figure 3: Graphs of $K=e^{-|x|^2}$ and $K_{N+1}$ with $N=10$ (top panels), the distributions of $\{\alpha_j\}_{1\le j\le N+1}$ (bottom panels) by using the Lagrange polynomial with the Chebyshev nodes. The graph of $K$ and $K_{N+1}$ are shown by the solid line and the dashed line, respectively. The numerical examples correspond to the results with $n=1$ (left), $n=2$ (middle), and $n=3$ (right), respectively.
  • Figure 4: Graphs of $K(x)=(1-|x|)\chi_{B(1)}(x)$ and $K_{N+1}$ with $N=19$ (top panels), the distributions of $\{\alpha_j\}_{1\le j\le N+1}$ (bottom panels) by using the Lagrange polynomial with the Chebyshev nodes. The graph of $K$ and $K_{N+1}$ are shown by the solid line and the dashed line, respectively. The numerical examples correspond to the results with $n=1$ (left), $n=2$ (middle), and $n=3$ (right), respectively.

Theorems & Definitions (56)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Lemma 1
  • Theorem 4
  • Theorem 5
  • Corollary 1
  • Remark 1
  • Theorem 6
  • Remark 2
  • ...and 46 more