Discontinuous stationary solutions to certain reaction-diffusion systems

TL;DR

This paper investigates reaction-diffusion systems coupled with ODEs, demonstrating that while both smooth and discontinuous stationary solutions exist, only the discontinuous solutions are stable, with implications for pattern formation.

Contribution

It identifies conditions under which discontinuous stationary solutions are stable in certain coupled reaction-diffusion systems with specific nonlinearities.

Findings

Discontinuous solutions can be stable in these systems.

Both smooth and discontinuous stationary solutions exist.

Stability is exclusive to discontinuous solutions.

Abstract

Systems consisting of a single ordinary differential equation coupled with one reaction-diffusion equation in a bounded domain and with the Neumann boundary conditions are studied in the case of particular nonlinearities from the Brusselator model, the Gray-Scott model, the Oregonator model and a certain predator-prey model. It is shown that the considered systems have the both smooth and discontinuous stationary solutions, however, only discontinuous ones can be stable.