On a Reversible Gray-Scott Type System from Energetic Variational Approach and Its Irreversible Limit

TL;DR

This paper introduces a reversible Gray-Scott type reaction-diffusion system derived via energetic variational principles, analyzes its well-posedness, and explores its limit to the classical irreversible model.

Contribution

It presents a novel reversible Gray-Scott model with an entropy structure and rigorously justifies its convergence to the irreversible system.

Findings

Established local well-posedness of the reversible system.

Proved global existence of solutions under small initial data.

Justified the limit from reversible to irreversible Gray-Scott model.

Abstract

Most of the previous studies on the well-known Gray-Scott model view it as an irreversible chemical reaction system. In this paper, we derive a four-species reaction-diffusion system using the energetic variational approach based on the law of mass action. This is a reversible Gray-Scott type model, which has a natural entropy structure. We establish the local well-posedness of this system, and justify the limit to the corresponding irreversible Gray-Scott type system as some backward coefficients tend to zero. Furthermore, under some smallness assumption on the initial data, we obtain the global-in-time existence of classical solutions of the reversible system.