Global Existence of Solutions for A Class of Nonlocal Reaction-Diffusion Systems and Their Diffusive Limit

TL;DR

The paper establishes global existence and uniqueness for an $m$-component nonlocal reaction-diffusion system on bounded domains under quasi-positivity and mass-control conditions, and derives $L^p$-boundedness that is independent of the nonlocal kernel. It introduces an $L^p$-energy framework with intermediate-sum structures to prove uniform bounds and a diffusive limit to a local reaction-diffusion system with diffusion coefficients $D_i = \frac{M d_i}{2n}$, where $M=\\int_{\mathbb{R}^n} |z|^2 \\psi(|z|) \,dz$. The results extend Laurencot-Walker by handling arbitrary $m$ and general nonlinearities, and they are complemented by numerical simulations that validate the diffusive limit and reveal qualitative differences between nonlocal and local diffusion, including mixed-diffusion scenarios with Neumann boundaries. Overall, the work provides a rigorous foundation for multi-component NRDEs with long-range interactions and offers practical numerical evidence of convergence to local models. These contributions advance the mathematical understanding of NRDEs and have potential implications for modeling complex systems with nonlocal interactions in biology and materials science.

Abstract

In this work, we study the global existence of solutions for a class of semilinear nonlocal reaction-diffusion systems with $m$ components on a bounded domain $Ω$ in $\mathbb{R}^n$ with smooth boundary. The initial data is assumed to be component-wise nonnegative and bounded, and the reaction vector field associated with the system is assumed to be quasi-positive and satisfy a generalized mass control condition. We obtain global existence and uniqueness of component-wise nonnegative solutions. With the additional assumption that the reaction vector field satisfies a linear intermediate sums condition, we employ an $L^p$ energy type functional to establish the uniform boundedness of solutions in $L^p(Ω)$ for all $2 \le p<\infty$ independent of the nonlocal diffusion operator for our system in $L^p$ space for $2 \le p < \infty$. This allows us to generalize a recent diffusive limit result of Laurencot and Walker \cite{laurenccot2023nonlocal}. We, also analyze a class of $m$ component reaction-diffusion systems in which some of the components diffuse nonlocally and the other components diffuse locally, where the latter components satisfy homogeneous Neumann boundary conditions. Under various assumptions, we establish global existence and uniqueness of componentwise nonnegative solutions by using duality arguments. Finally, we numerically verify our diffusive limit result. We also numerically solve the reaction-diffusion systems with a mixture of nonlocal and local diffusion and show the visual difference of its solutions with the system in which all components diffuse nonlocally.

Figures (7)

• Figure 1: Solutions of local Gray-Scott model for $a=0.25$, $b=0.080$, $d_1 = 0.1$, $d_2 = 0.01$.
• Figure 2: Solutions of nonlocal Gray-Scott model for $a=0.25$, $b=0.080$, $d_1 = 0.1$, $d_2 = 0.01$.
• Figure 3: Solutions of nonlocal Gray-Scott model for $j=1$, $a=0.25$, $b=0.080$, $d_1 = 0.1$, $d_2 = 0.01$.
• Figure 4: Solutions of nonlocal Gray-Scott model for $j=3$, $a=0.25$, $b=0.080$, $d_1 = 0.1$, $d_2 = 0.01$.
• Figure 5: Solutions of nonlocal Gray-Scott model for $j=7$, $\epsilon=0.81$, $a=0.25$, $b=0.080$, $d_1 = 0.1$, $d_2 = 0.01$.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4