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

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## Abstract

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.

## Keywords

Advection-reaction equation Diffusion equation Asymptotic stability Oxygen transport Alveolar capillary## Mathematics Subject Classification

35B40 35Q80## Notes

### Acknowledgments

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.

## References

- 1.Bertsch, M., Hilhorst, D., Izuhara, H., Mimura, M.: A nonlinear parabolic-hyperbolic system for constant inhibition of cell-growth. Differ. Equ. Appl.
**4**, 137–157 (2012)MathSciNetzbMATHGoogle Scholar - 2.Cui, S.: Asymptotic stability of the stationary solution for a parabolic-hyperbolic free boundary problem modeling tumor growth. SIAM J. Math. Anal.
**45**, 2870–2893 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Cui, S., Wei, X.: Global existence for a parabolic-hyperbolic free boundary problem modeling tumor growth. Acta Math. Appl. Sin. (English Ser.)
**21**, 597–614 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Friedman, A.: Partial Differential Equations of Parabolic Type. Krieger, Malabar (1983)zbMATHGoogle Scholar
- 5.Hale, J.K.: Ordinary Differential Equations. Krieger, Malabar (1980)zbMATHGoogle Scholar
- 6.Jimbo, S., Morita, Y.: Lyapunov function and spectrum comparison for a reaction-diffusion system with mass conservation. J. Differ. Equ.
**255**, 1657–1683 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 7.John, F.: Partial Differential Equations, 4th edn. Springer, New York (1982)CrossRefGoogle Scholar
- 8.Keener, J., Sneyd, J.: Mathematical Physiology II: Systems Physiology, 2nd edn. Springer, New York (2009)Google Scholar
- 9.Maury, B.: The Respiratory System in Equations, MS&A, vol. 7. Springer, Milan (2013)CrossRefGoogle Scholar
- 10.Ottesen, J.T., Olufsen, M.S., Larsen, J.K.: Applied Mathematical Models in Human Physiology. Monographs on Mathematical Modeling and Computation. SIAM, Philadelphia (2004)CrossRefGoogle Scholar
- 11.Smith, H.L.: Monotone Dynamical Systems : An Introduction to the Theory of Competitive and Cooperative Systems. American Mathematical Society, Providence (2008)CrossRefGoogle Scholar
- 12.Toro, E.F., Siviglia, A.: Simplified blood flow model with discontinuous vessel properties, modeling of physiological flows. In: Ambrosi, D., Quarteroni, A., Rozza, G. (eds.) MS&A, vol. 5, pp. 19–39. Springer, Milan (2012)Google Scholar