Convergence to equilibrium for a degenerate triangular reaction-diffusion system

TL;DR

This paper investigates the long-term behavior of a reaction-diffusion system with degenerate and triangular nonlinearities, proving convergence to equilibrium under specific conditions across various spatial dimensions.

Contribution

It establishes convergence to equilibrium for a degenerate reaction-diffusion system with triangular non-linearity, including cases with zero diffusion coefficients, in multiple dimensions.

Findings

Proves convergence to equilibrium in dimensions 1, 2, and 3.

Extends results to higher dimensions under closeness conditions on diffusion coefficients.

Provides explicit constants in decay estimates.

Abstract

In this article we study a reaction diffusion system with $m$ unknown concentration. The non-linearity in our study comes from an underlying reversible chemical reaction and triangular in nature. Our objective is to understand the large time behaviour of solution where there are degeneracies. In particular we treat those cases when one of the diffusion coefficient is zero and others are strictly positive. We prove convergence to equilibrium type of results under some condition on stoichiometric coefficients in dimension $1$,$2$ and $3$ in correspondence with the existence of classical solution. For dimension greater than 3 we prove similar result under certain closeness condition on the non-zero diffusion coefficients and with the same condition imposed on stoichiometric coefficients. All the constant occurs in the decay estimates are explicit.