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December  2014, 9(4): 655-668. doi: 10.3934/nhm.2014.9.655

A one dimensional free boundary problem for adsorption phenomena

 1 Division of General Education, Nagaoka National College of Technology, 888, Nishikatakai, Nagaoka, Niigata, 940-8532, Japan 2 Japan Woman's University, Department of Mathematics and Physical Sciences, Faculty of Science, 2-8-1 Mejirodai, Bunkyo-ku, Tokyo 112-8681, Japan 3 Department of Mathematics, Faculty of Science and Technology, Meijo University, 1-501 Shiogamaguchi, Tenpaku-ku, Nagoya, 468-8502 4 Department of Mathematics, Faculty of Education, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522

Received  February 2014 Revised  September 2014 Published  December 2014

In this paper we deal with a one-dimensional free boundary problem, which is a mathematical model for an adsorption phenomena appearing in concrete carbonation process. This model was proposed in line of previous studies of three dimensional concrete carbonation process. The main result in this paper is concerned with the existence and uniqueness of a time-local solution to the free boundary problem. This result will be obtained by means of the abstract theory of nonlinear evolution equations and Banach's fixed point theorem, and especially, the maximum principle applied to our problem will play a very important role to obtain the uniform estimate to approximate solutions.
Citation: Naoki Sato, Toyohiko Aiki, Yusuke Murase, Ken Shirakawa. A one dimensional free boundary problem for adsorption phenomena. Networks and Heterogeneous Media, 2014, 9 (4) : 655-668. doi: 10.3934/nhm.2014.9.655
References:
 [1] T. Aiki and K. Kumazaki, Well-posedness of a mathematical model for moisture transport appearing in concrete carbonation process, Adv. Math. Sci. Appl., 21 (2011), 361-381. [2] T. Aiki and K. Kumazaki, Mathematical model for hysteresis phenomenon in moisture transport of concrete carbonation process, Physica B, 407 (2012), 1424-1426. doi: 10.1016/j.physb.2011.10.016. [3] T. Aiki and K. Kumazaki, Mathematical modeling of concrete carbonation process with hysteresis effect. Sūrikaisekikenkyūsho Kōkyūroku, 1792 (2012), 98-107. [4] T. Aiki and A. Muntean, Existence and uniqueness of solutions to a mathematical model predicting service life of concrete structures, Adv. Math. Sci. Appl., 19 (2009), 109-129. [5] T. Aiki and A. Muntean, Large time behavior of solutions to concrete carbonation problem, Communications on Pure and Applied Analysis, 9 (2010), 1117-1129. doi: 10.3934/cpaa.2010.9.1117. [6] T. Aiki and A. Muntean, A free-boundary problem for concrete carbonation: Rigorous justification of $\sqrtt$-law of propagation, Interfaces and Free Bound., 15 (2013), 167-180. doi: 10.4171/IFB/299. [7] T. Aiki and A. Muntean, Large-time asymptotics of moving-reaction interfaces involving nonlinear Henry's law and time-dependent Dirichlet data, Nonlinear Anal., 93 (2013), 3-14. doi: 10.1016/j.na.2013.07.002. [8] T. Aiki, Y. Murase, N. Sato and K. Shirawaka, A mathematical model for a hysteresis appearing in adsorption phenomena, Sūrikaisekikenkyūsho Kōkyūroku, 1856 (2013), 1-12. [9] A. Fasano and M. Primicerio, Free boundary problems for nonlinear parabolic equations with nonlinear free boundary conditions, J. Math. Anal. Appl., 72 (1979), 247-273. doi: 10.1016/0022-247X(79)90287-7. [10] A. Fasano and M. Primicerio, General free-boundary problems for the heat equation. II, J. Math. Anal. Appl., 58 (1977), 202-231. doi: 10.1016/0022-247X(77)90239-6. [11] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87. [12] K. Maekawa, R. Chaube and T. Kishi, Modeling of Concrete Performance, Taylor and Francis, 1999. [13] K. Maekawa, T. Ishida and T. Kishi, Multi-scale modeling of concrete performance, Journal of Advanced Concrete Technology, 1 (2003), 91-126. doi: 10.3151/jact.1.91. [14] A. Muntean and M. Böhm, A moving-boundary problem for concrete carbonation: Global existence and uniqueness of solutions, Journal of Mathematical Analysis and Applications, 350 (2009), 234-251. doi: 10.1016/j.jmaa.2008.09.044.

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References:
 [1] T. Aiki and K. Kumazaki, Well-posedness of a mathematical model for moisture transport appearing in concrete carbonation process, Adv. Math. Sci. Appl., 21 (2011), 361-381. [2] T. Aiki and K. Kumazaki, Mathematical model for hysteresis phenomenon in moisture transport of concrete carbonation process, Physica B, 407 (2012), 1424-1426. doi: 10.1016/j.physb.2011.10.016. [3] T. Aiki and K. Kumazaki, Mathematical modeling of concrete carbonation process with hysteresis effect. Sūrikaisekikenkyūsho Kōkyūroku, 1792 (2012), 98-107. [4] T. Aiki and A. Muntean, Existence and uniqueness of solutions to a mathematical model predicting service life of concrete structures, Adv. Math. Sci. Appl., 19 (2009), 109-129. [5] T. Aiki and A. Muntean, Large time behavior of solutions to concrete carbonation problem, Communications on Pure and Applied Analysis, 9 (2010), 1117-1129. doi: 10.3934/cpaa.2010.9.1117. [6] T. Aiki and A. Muntean, A free-boundary problem for concrete carbonation: Rigorous justification of $\sqrtt$-law of propagation, Interfaces and Free Bound., 15 (2013), 167-180. doi: 10.4171/IFB/299. [7] T. Aiki and A. Muntean, Large-time asymptotics of moving-reaction interfaces involving nonlinear Henry's law and time-dependent Dirichlet data, Nonlinear Anal., 93 (2013), 3-14. doi: 10.1016/j.na.2013.07.002. [8] T. Aiki, Y. Murase, N. Sato and K. Shirawaka, A mathematical model for a hysteresis appearing in adsorption phenomena, Sūrikaisekikenkyūsho Kōkyūroku, 1856 (2013), 1-12. [9] A. Fasano and M. Primicerio, Free boundary problems for nonlinear parabolic equations with nonlinear free boundary conditions, J. Math. Anal. Appl., 72 (1979), 247-273. doi: 10.1016/0022-247X(79)90287-7. [10] A. Fasano and M. Primicerio, General free-boundary problems for the heat equation. II, J. Math. Anal. Appl., 58 (1977), 202-231. doi: 10.1016/0022-247X(77)90239-6. [11] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87. [12] K. Maekawa, R. Chaube and T. Kishi, Modeling of Concrete Performance, Taylor and Francis, 1999. [13] K. Maekawa, T. Ishida and T. Kishi, Multi-scale modeling of concrete performance, Journal of Advanced Concrete Technology, 1 (2003), 91-126. doi: 10.3151/jact.1.91. [14] A. Muntean and M. Böhm, A moving-boundary problem for concrete carbonation: Global existence and uniqueness of solutions, Journal of Mathematical Analysis and Applications, 350 (2009), 234-251. doi: 10.1016/j.jmaa.2008.09.044.
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