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

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.