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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 & 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.  Google Scholar [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.  Google Scholar [3] T. Aiki and K. Kumazaki, Mathematical modeling of concrete carbonation process with hysteresis effect. Sūrikaisekikenkyūsho Kōkyūroku, 1792 (2012), 98-107. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [11] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87. Google Scholar [12] K. Maekawa, R. Chaube and T. Kishi, Modeling of Concrete Performance, Taylor and Francis, 1999. Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [3] T. Aiki and K. Kumazaki, Mathematical modeling of concrete carbonation process with hysteresis effect. Sūrikaisekikenkyūsho Kōkyūroku, 1792 (2012), 98-107. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [11] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87. Google Scholar [12] K. Maekawa, R. Chaube and T. Kishi, Modeling of Concrete Performance, Taylor and Francis, 1999. Google Scholar [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.  Google Scholar [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.  Google Scholar
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