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December  2014, 9(4): 683-707. doi: 10.3934/nhm.2014.9.683

## Uniqueness of solutions to a mathematical model describing moisture transport in concrete materials

 1 Department of Mathematical and Physical Sciences, Faculty of Science, Japan Women's University, 2-8-1 Mejirodai, Bunkyo-ku, Tokyo 112-8681 2 Natural and Physical Sciences, Tomakomai National College of Technology, 443, Nishikioka, Tomakomai-shi, Hokkaido, 059-1275

Received  June 2014 Revised  August 2014 Published  December 2014

When dealing with concrete materials it is always a big issue how to deal with the moisture transport. Here, we consider a mathematical model for moisture transport, which is given as a system consisting of the diffusion equation for moisture and of the ordinary differential equation which describes a hysteresis operator. In [3] we already proved the existence of a time global solution of an initial boundary value problem of this system, however, the uniqueness is obtained only for one dimensional domains. The main purpose of this paper is to establish the uniqueness of a solution of our problem in three dimensional domains under the assumption of the smooth boundary and initial data.
Citation: Toyohiko Aiki, Kota Kumazaki. Uniqueness of solutions to a mathematical model describing moisture transport in concrete materials. Networks and Heterogeneous Media, 2014, 9 (4) : 683-707. doi: 10.3934/nhm.2014.9.683
##### References:
 [1] T. Aiki and K. Kumazaki, Mathematical model for hysteresis phenomenon in moisture transport in concrete carbonation process, Phys. B, 407 (2012), 1424-1426. doi: 10.1016/j.physb.2011.10.016. [2] T. Aiki and K. Kumazaki, Mathematical modelling of concrete carbonation process with hysteresis effect, RIMS, Kyoto Univ., sūrikaisekikenkyūsho, kōkyūuroku, 1792 (2012), 99-107. [3] 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. [4] B. Bary and A. Sellier, Coupled moisture-Carbon dioxide-Calcium transfer model for carbonation of concrete, Cem. Concr. Res., 34 (2004), 1859-1872. doi: 10.1016/j.cemconres.2004.01.025. [5] O. V. Besov, V. P. ll'in and S. M. Nikol'ski, Integral Representations of Functions and Embedding Theorems, Vol. II. Scripta Series in Mathematics. Edited by Mitchell H. Taibleson. V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London, 1979. [6] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer-Verlag, 1996. doi: 10.1007/978-1-4612-4048-8. [7] H. Derluyn, D. Derome, J. Carmeliet, E. Stora and R. Barbarulo, Hysteric moisture behavior of concrete: Modelling analysis, Cem. Concr. Res., 42 (2012), 1379-1388. [8] P. Colli, N. Kenmochi and M. Kubo, A phase field model with temperature dependent constraint, J. Math. Anal. Appl., 256 (2001), 668-685. doi: 10.1006/jmaa.2000.7338. [9] N. Kenmochi, T. Koyama and G. H. Meyer, Parabolic PDEs with hysteresis and quasivariational inequalities, Nonlinear Anal., 34 (1998), 665-686. doi: 10.1016/S0362-546X(97)00592-0. [10] O. A. Ladyženskaja, V. A. Solonnilov and N. N. Uralćeva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Society, Providence RI, 1967. [11] O. A. Ladyženskaja and N. N. Ural'ceva, Équations Aux Dérivées Partielles de Type Elliptique, Dunod, Paris, 1968. [12] J. Nečas, Les Methodes Directes en Theorie des Equations Elliptiques, Academia, Praha, and Masson et Cie Editeurs, Paris, 1967. [13] A. Visintin, Differential Models of Hysteresis, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-11557-2.

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##### References:
 [1] T. Aiki and K. Kumazaki, Mathematical model for hysteresis phenomenon in moisture transport in concrete carbonation process, Phys. B, 407 (2012), 1424-1426. doi: 10.1016/j.physb.2011.10.016. [2] T. Aiki and K. Kumazaki, Mathematical modelling of concrete carbonation process with hysteresis effect, RIMS, Kyoto Univ., sūrikaisekikenkyūsho, kōkyūuroku, 1792 (2012), 99-107. [3] 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. [4] B. Bary and A. Sellier, Coupled moisture-Carbon dioxide-Calcium transfer model for carbonation of concrete, Cem. Concr. Res., 34 (2004), 1859-1872. doi: 10.1016/j.cemconres.2004.01.025. [5] O. V. Besov, V. P. ll'in and S. M. Nikol'ski, Integral Representations of Functions and Embedding Theorems, Vol. II. Scripta Series in Mathematics. Edited by Mitchell H. Taibleson. V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London, 1979. [6] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer-Verlag, 1996. doi: 10.1007/978-1-4612-4048-8. [7] H. Derluyn, D. Derome, J. Carmeliet, E. Stora and R. Barbarulo, Hysteric moisture behavior of concrete: Modelling analysis, Cem. Concr. Res., 42 (2012), 1379-1388. [8] P. Colli, N. Kenmochi and M. Kubo, A phase field model with temperature dependent constraint, J. Math. Anal. Appl., 256 (2001), 668-685. doi: 10.1006/jmaa.2000.7338. [9] N. Kenmochi, T. Koyama and G. H. Meyer, Parabolic PDEs with hysteresis and quasivariational inequalities, Nonlinear Anal., 34 (1998), 665-686. doi: 10.1016/S0362-546X(97)00592-0. [10] O. A. Ladyženskaja, V. A. Solonnilov and N. N. Uralćeva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Society, Providence RI, 1967. [11] O. A. Ladyženskaja and N. N. Ural'ceva, Équations Aux Dérivées Partielles de Type Elliptique, Dunod, Paris, 1968. [12] J. Nečas, Les Methodes Directes en Theorie des Equations Elliptiques, Academia, Praha, and Masson et Cie Editeurs, Paris, 1967. [13] A. Visintin, Differential Models of Hysteresis, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-11557-2.
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