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October  2011, 29(4): 1345-1365. doi: 10.3934/dcds.2011.29.1345

## On uniqueness of a weak solution of one-dimensional concrete carbonation problem

 1 Department of Mathematics, Faculty of Education, Gifu University, Yanagido 1-1, Gifu, 501-1193 2 CASA - Centre for Analysis, Scientific computing and Applications, Department of Mathematics and Computer Science, Institute of Complex Molecular Systems, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven

Received  January 2010 Revised  August 2010 Published  December 2010

In our previous works we studied a one-dimensional free-boundary model related to the aggressive penetration of gaseous carbon dioxide in unsaturated concrete. Essentially, global existence and uniqueness of weak solutions to the model were obtained when the initial functions are bounded on the domain. In this paper we investigate the well-posedness of the problem for the case when the initial functions belong to a $L^2-$ class. Specifically, the uniqueness of weak solutions is proved by applying the dual equation method.
Citation: Toyohiko Aiki, Adrian Muntean. On uniqueness of a weak solution of one-dimensional concrete carbonation problem. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1345-1365. doi: 10.3934/dcds.2011.29.1345
##### References:
 [1] T. Aiki, Weak solutions for Falk's model of shape memory alloys, Math. Methods Appl. Sci., 23 (2000), 299-319. doi: 10.1002/(SICI)1099-1476(20000310)23:4<299::AID-MMA115>3.0.CO;2-D. [2] T. Aiki, Uniqueness for multi-dimensional Stefan problems with nonlinear boundary conditions described by maximal monotone operators, Differential Integral Equations, 15 (2002), 973-1008. [3] 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. [4] T. Aiki and A. Muntean, Large time behavior of solutions to a concrete carbonation problem, Commun. Pure Appl. Anal., 9 (2010), 1117-1129. doi: 10.3934/cpaa.2010.9.1117. [5] T. Aiki and A. Muntean, Mathematical treatment of concrete carbonation process, in "Current Advances in Nonlinear Analysis and Related Topics," GAKUTO Internat. Ser. Math. Sci. Appl., 32, Gakkotosho, Tokyo, (2010), 231-238. [6] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type," Transl. Math. Monograph, 23, Amer. Math. Soc., Providence, R. I., 1968. [7] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, Paris, 1969. [8] A. Muntean, "A Moving-Boundary Problem: Modeling, Analysis and Simulation of Concrete Carbonation," Ph.D. Thesis, Faculty of Mathematics, University of Bremen, Germany, Cuvillier Verlag, Göttingen, 2006. [9] 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. [10] M. Niezgódka and I. Pawlow, A generalized Stefan problem in several space variables, Applied Math. Opt., 9 (1983), 193-224. doi: 10.1007/BF01460125.

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##### References:
 [1] T. Aiki, Weak solutions for Falk's model of shape memory alloys, Math. Methods Appl. Sci., 23 (2000), 299-319. doi: 10.1002/(SICI)1099-1476(20000310)23:4<299::AID-MMA115>3.0.CO;2-D. [2] T. Aiki, Uniqueness for multi-dimensional Stefan problems with nonlinear boundary conditions described by maximal monotone operators, Differential Integral Equations, 15 (2002), 973-1008. [3] 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. [4] T. Aiki and A. Muntean, Large time behavior of solutions to a concrete carbonation problem, Commun. Pure Appl. Anal., 9 (2010), 1117-1129. doi: 10.3934/cpaa.2010.9.1117. [5] T. Aiki and A. Muntean, Mathematical treatment of concrete carbonation process, in "Current Advances in Nonlinear Analysis and Related Topics," GAKUTO Internat. Ser. Math. Sci. Appl., 32, Gakkotosho, Tokyo, (2010), 231-238. [6] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type," Transl. Math. Monograph, 23, Amer. Math. Soc., Providence, R. I., 1968. [7] J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, Paris, 1969. [8] A. Muntean, "A Moving-Boundary Problem: Modeling, Analysis and Simulation of Concrete Carbonation," Ph.D. Thesis, Faculty of Mathematics, University of Bremen, Germany, Cuvillier Verlag, Göttingen, 2006. [9] 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. [10] M. Niezgódka and I. Pawlow, A generalized Stefan problem in several space variables, Applied Math. Opt., 9 (1983), 193-224. doi: 10.1007/BF01460125.
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