On a constrained reaction-diffusion system related to multiphase problems

TL;DR

This paper analyzes a reaction-diffusion system within the Gibbs simplex, characterizing solutions and Lagrange multipliers through variational inequalities, and investigates phase stability and properties of the two obstacles evolution problem.

Contribution

It introduces a novel approach to solving and characterizing a constrained reaction-diffusion system using variational inequalities and phase stability analysis.

Findings

Characterization of Lagrange multipliers in the system

Approximation of the N-system with characteristic functions

Conditions for phase stability and non-degeneracy

Abstract

We solve and characterize the Lagrange multipliers of a reaction-diffusion system in the Gibbs simplex of R^{N+1} by considering strong solutions of a system of parabolic variational inequalities in R^N. Exploring properties of the two obstacles evolution problem, we obtain and approximate a N-system involving the characteristic functions of the saturated and/or degenerated phases in the nonlinear reaction terms. We also show continuous dependence results and we establish sufficient conditions of non-degeneracy for the stability of those phase subregions.