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

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 $\Omega$ 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 $L^p(\Omega)$ 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.