Well-Posedness for Fractional Reaction-Diffusion Systems with Mass Dissipation in $\mathbb R^N$

TL;DR

This paper establishes the global existence and boundedness of solutions for fractional reaction-diffusion systems with mass dissipation in unbounded domains, extending classical results to non-local diffusion cases.

Contribution

It introduces new methods to prove global solutions for super-quadratic nonlinearities in fractional reaction-diffusion systems with mass dissipation.

Findings

Global bounded solutions exist for quadratic nonlinearities.

Duality methods are adapted for super-quadratic cases.

Results extend reaction-diffusion theory to fractional, non-local diffusion.

Abstract

The global existence of bounded solutions to reaction-diffusion systems with fractional diffusion in the whole space $\mathbb R^N$ is investigated. The systems are assumed to preserve the non-negativity of initial data and to dissipate total mass. We first show that if the nonlinearities are at most quadratic then there exists a unique global bounded solution regardless of the fractional order. This is done by combining a regularizing effect of the fractional diffusion operator and the H\"older continuity of a non-local inhomogeneous parabolic equation. When the nonlinearities might be super-quadratic, but satisfy some intermediate sum conditions, we prove the global existence of bounded solutions by adapting the well-known duality methods to the case of fractional diffusion. In this case, the order of the intermediate sum conditions depends on the fractional order. These results extend the existing theory for mass dissipated reaction-diffusion systems to the case of non-local diffusion and unbounded domains.