Maximum principle and local stability for a class of coupled nonlinear thermo--reaction--phase systems

TL;DR

This paper analyzes a coupled nonlinear thermo-reaction-phase PDE system, establishing maximum principles for positivity, invariance of physical variables, and local stability of stationary states.

Contribution

It introduces a maximum principle for positivity, invariance of variables, and proves local asymptotic stability in a coupled thermo-reaction-phase model.

Findings

Positivity of temperature is maintained over time.

Variables remain within physically admissible bounds.

Homogeneous stationary states are locally asymptotically stable.

Abstract

We study a nonlinear coupled system of partial differential equations arising from thermo--reaction--phase models. The system combines a heat diffusion equation, temperature-dependent chemical reactions of Arrhenius type, and a phase variable, and is formulated as a strongly coupled parabolic problem with homogeneous Neumann boundary conditions. We first establish a maximum principle ensuring the positivity of the temperature on a suitable time interval, as well as the invariance of the physically admissible domain. In particular, we prove that the internal variables remain in the interval [0,1]. We then analyse the asymptotic behaviour of the system in the free regime, that is, in the absence of external forcing. By introducing a relative energy functional and exploiting the structure of the coupling terms, we obtain local asymptotic stability of a homogeneous stationary state. The model belongs to a broader class of coupled diffusion--reaction--phase systems.