A weakly coupled system of advection-reaction and diffusion equations in physiological gas transport

We deal with a simple model for oxygen transport in alveolar capillaries with exchange of oxygen between the capillaries and alveoli. This model is described by a weakly coupled three-component system of advection-reaction equations in capillaries and a linear diffusion equation in alveoli. We consider the equations in a bounded interval with appropriate boundary conditions. The goal of this article is to show that a steady state solution of the equations is asymptotically stable. To this end we first establish the existence of a unique solution for an initial-boundary value problem of the equations. Then we show the existence of a steady state solution. Finally we prove the main result on the asymptotic stability of the steady state with an exponential convergence rate. The proof can be done by using energy estimates for a large coupling constant.