Global Existence and Diffusive Limits for a Class of Nonlocal Reaction-Diffusion Systems

We study a class of semilinear reaction-diffusion systems with nonlocal diffusion 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 is assumed to be quasi-positive and satisfy a generalized mass control condition.

We obtain global existence and uniqueness of component-wise nonnegative solutions, and when the reaction vector field satisfies a linear intermediate sum condition, we establish the uniform boundedness of solutions in $L^p(\Omega)$ for all $2 \le p<\infty$ on bounded time intervals independent of the kernel of the nonlocal diffusion operator.

This allows us to generalize a recent diffusive limit result of Laurencot and Walker \cite{laurencot2023nonlocal}.

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, and establish both global existence and a diffusive limit.