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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 & Continuous Dynamical Systems - A, 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.  doi: 10.1002/(SICI)1099-1476(20000310)23:4<299::AID-MMA115>3.0.CO;2-D.  Google Scholar

[2]

T. Aiki, Uniqueness for multi-dimensional Stefan problems with nonlinear boundary conditions described by maximal monotone operators,, Differential Integral Equations, 15 (2002), 973.   Google Scholar

[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.   Google Scholar

[4]

T. Aiki and A. Muntean, Large time behavior of solutions to a concrete carbonation problem,, Commun. Pure Appl. Anal., 9 (2010), 1117.  doi: 10.3934/cpaa.2010.9.1117.  Google Scholar

[5]

T. Aiki and A. Muntean, Mathematical treatment of concrete carbonation process,, in, 32 (2010), 231.   Google Scholar

[6]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type,", Transl. Math. Monograph, 23 (1968).   Google Scholar

[7]

J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", Dunod, (1969).   Google Scholar

[8]

A. Muntean, "A Moving-Boundary Problem: Modeling, Analysis and Simulation of Concrete Carbonation,", Ph.D. Thesis, (2006).   Google Scholar

[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.  doi: 10.1016/j.jmaa.2008.09.044.  Google Scholar

[10]

M. Niezgódka and I. Pawlow, A generalized Stefan problem in several space variables,, Applied Math. Opt., 9 (1983), 193.  doi: 10.1007/BF01460125.  Google Scholar

show all references

References:
[1]

T. Aiki, Weak solutions for Falk's model of shape memory alloys,, Math. Methods Appl. Sci., 23 (2000), 299.  doi: 10.1002/(SICI)1099-1476(20000310)23:4<299::AID-MMA115>3.0.CO;2-D.  Google Scholar

[2]

T. Aiki, Uniqueness for multi-dimensional Stefan problems with nonlinear boundary conditions described by maximal monotone operators,, Differential Integral Equations, 15 (2002), 973.   Google Scholar

[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.   Google Scholar

[4]

T. Aiki and A. Muntean, Large time behavior of solutions to a concrete carbonation problem,, Commun. Pure Appl. Anal., 9 (2010), 1117.  doi: 10.3934/cpaa.2010.9.1117.  Google Scholar

[5]

T. Aiki and A. Muntean, Mathematical treatment of concrete carbonation process,, in, 32 (2010), 231.   Google Scholar

[6]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type,", Transl. Math. Monograph, 23 (1968).   Google Scholar

[7]

J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", Dunod, (1969).   Google Scholar

[8]

A. Muntean, "A Moving-Boundary Problem: Modeling, Analysis and Simulation of Concrete Carbonation,", Ph.D. Thesis, (2006).   Google Scholar

[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.  doi: 10.1016/j.jmaa.2008.09.044.  Google Scholar

[10]

M. Niezgódka and I. Pawlow, A generalized Stefan problem in several space variables,, Applied Math. Opt., 9 (1983), 193.  doi: 10.1007/BF01460125.  Google Scholar

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