A weakly coupled system of advection-reaction and diffusion equations in physiological gas transport
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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.
KeywordsAdvection-reaction equation Diffusion equation Asymptotic stability Oxygen transport Alveolar capillary
Mathematics Subject Classification35B40 35Q80
The authors would like to express their thanks to Professor Masaharu Nagayama for his useful comment on the chemical model for hemoglobin and oxygen. They also would like to thank the referees for useful comments for the revisions. The first author was partially supported by JSPS KAKENHI Grant Number 26287025, 24654044, 26247013.
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