«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

## Existence of periodic solution to one dimensional free boundary problems for adsorption phenomena

Author(s)
T. Aiki, N. Sato
Abstract
In this paper we consider a drying and wetting process in porous medium to create a mathematical model for concrete carbonation. The process is assumed to be characterized by the growth of the air zone and a diffusion of moisture in the air zone. Under the assumption we proposed a one-dimensional free boundary problem describing adsorption phenomena in a porous medium. The free boundary problem it to find a curve representing the air zone and the relative humidity of the air zone. For the problem we also established existence, uniqueness and a large time behavior of solutions. Here, by improving the method for uniform estimates we can show the existence of a periodic solution of the problem. Also, the extension method is applied in the proof. This idea is quite important and new since the value of the humidity on the free boundary is unknown.

Toyohiko Aiki, Dr. Sci. (Phys.–Math.), Prof., Department of Mathematics, Faculty of Science, Japan Women’s University, 2-8-1 Mejirodai, Bunkyo-ku, Tokyo, 112-8681, Japan, e-mail: aikit@fc.jwu.ac.jp

Naoki Sato, Dr. Sci. (Phys.–Math.), Assoc. Prof., Division of General Education, National Institute of Technology, Nagaoka College, 888, Nishikatakai, Nagaoka, Niigata, 940-8532, Japan, e-mail: naoki@nagaoka-ct.ac.jp

For citation

Aiki T., Sato N. Existence of Periodic Solution of One Dimensional Free Boundary Problem for Adsorption Phenomena. The Bulletin of Irkutsk State University. Series Mathematics, 2018, vol. 25, pp. 3-18. https://doi.org/10.26516/1997-7670.2018.25.3

Keywords
free boundary problem, periodic solution, fixed point argument
UDC
518.517
MSC
35R35, 35K60, 35B40
DOI
https://doi.org/10.26516/1997-7670.2018.25.3
References

1. Aiki T. Periodic stability of solutions to two-phase Stefan problems with nonlinear boundary condition. Nonlinear Anal. TMA., 1994, vol. 22, pp. 1445–1474.

2. Aiki T., Kumazaki K. Mathematical model for hysteresis phenomenon in moisture transport of concrete carbonation process. Physica B, 2012, vol. 407, pp. 1424– 1426.

3. Aiki T., Kumazaki K. Well-posedness of a mathematical model for moisture transport appearing in concrete carbonation process. Adv. Math. Sci. Appl., 2011, vol. 21, pp. 361–381.

4. Aiki T., Kumazaki K., Murase Y., Sato N. A two-scale model for concrete carbonation process in a three dimensional domain. Surikaisekikenkyusho Kokyuroku, 2016, no. 1997, pp. 133–139.

5. Aiki T., Murase Y., Sato N., Shirakawa K. A mathematical model for a hysteresis appearing in adsorption phenomena. Surikaisekikenkyusho Kokyuroku, 2013, no. 1856, pp. 1–11.

6. Aiki T., Murase Y. On a large time behavior of a solution to a one-dimensional free boundary problem for adsorption phenomena. J. Math. Anal. Appl., 2017, vol. 445, pp. 837–854.

7. Damlamian A., Kenmochi N. Periodicity and almost periodicity of solutions to a multi-phase Stefan Problem in several space variables, Nonlinear Anal. TMA., 1988, vol. 12, pp. 921–934.

8. Kumazaki K. Continuous dependence of a solution of a free boundary problem describing adsorption phenomenon for a given data. Adv. Math. Sci. Appl., 2016, vol. 25, pp. 289–305.

9. Kumazaki K. Measurabllity of a solution of a free boundary problem describing adsorption phenomenon. Adv. Math. Sci. Appl., 2017, vol. 26, pp. 19–27.

10. Kumazaki K., Aiki T. Uniqueness of a solution for some parabolic type equation with hysteresis in three dimensions. Networks and Heterogeneous Media, 2014, vol. 9, pp. 683–707.

11. Kumazaki K., Aiki T., Sato N., Murase Y. Multiscale model for moisture transport with adsorption phenomenon in concrete materials. Appl. Anal., 2018, vol. 97, pp. 41–54.

12. Sato N., Aiki T., Murase Y., Shirakawa K. A one dimensional free boundary problem for adsorption phenomena. Netw. Heterog. Media, 2014, vol. 9, pp. 655–668.