A functional inequalities approach for the field-road diffusion model with (symmetric) nonlinear exchanges

TL;DR

This paper introduces a new functional inequalities method to analyze the field-road diffusion model, demonstrating exponential decay and convergence to equilibrium in a coupled PDE system with nonlinear exchanges.

Contribution

It presents a novel functional inequalities approach to prove exponential decay in a coupled PDE model with nonlinear exchange terms.

Findings

Proves exponential decay of solutions towards stationary state

Establishes convergence using a new functional inequalities method

Applicable to models with nonlinear exchange terms

Abstract

In this note, we consider the so-called field-road diffusion model in a bounded domain, consisting of two parabolic PDEs posed on sets of different dimensions and coupled through (symmetric) nonlinear exchange terms. We propose a new and rather direct functional inequalities approach to prove the exponential decay of a relative entropy, and thus the convergence of the solution towards the stationary state selected by the total mass of the initial datum.