• Previous Article
    A biharmonic-modified forward time stepping method for fourth order nonlinear diffusion equations
  • DCDS Home
  • This Issue
  • Next Article
    Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields
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.

show all references

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.

[1]

Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003

[2]

Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087

[3]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033

[4]

Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763

[5]

Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10

[6]

Yang Zhang. A free boundary problem of the cancer invasion. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1323-1343. doi: 10.3934/dcdsb.2021092

[7]

Yu-Feng Sun, Zheng Zeng, Jie Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 157-164. doi: 10.3934/naco.2019045

[8]

Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431

[9]

Aram L. Karakhanyan. Lipschitz continuity of free boundary in the continuous casting problem with divergence form elliptic equation. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 261-277. doi: 10.3934/dcds.2016.36.261

[10]

Sergey Degtyarev. Classical solvability of the multidimensional free boundary problem for the thin film equation with quadratic mobility in the case of partial wetting. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3625-3699. doi: 10.3934/dcds.2017156

[11]

Yuki Kaneko, Hiroshi Matsuzawa, Yoshio Yamada. A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions I : Classification of asymptotic behavior. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2719-2745. doi: 10.3934/dcds.2021209

[12]

Mingxin Wang. Existence and uniqueness of solutions of free boundary problems in heterogeneous environments. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 415-421. doi: 10.3934/dcdsb.2018179

[13]

Mingxin Wang. Erratum: Existence and uniqueness of solutions of free boundary problems in heterogeneous environments. Discrete and Continuous Dynamical Systems - B, 2022, 27 (9) : 5179-5180. doi: 10.3934/dcdsb.2021269

[14]

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

[15]

Yongzhi Xu. A free boundary problem model of ductal carcinoma in situ. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 337-348. doi: 10.3934/dcdsb.2004.4.337

[16]

Anna Lisa Amadori. Contour enhancement via a singular free boundary problem. Conference Publications, 2007, 2007 (Special) : 44-53. doi: 10.3934/proc.2007.2007.44

[17]

Shihe Xu. Analysis of a delayed free boundary problem for tumor growth. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 293-308. doi: 10.3934/dcdsb.2011.15.293

[18]

Igor Kukavica, Amjad Tuffaha. On the 2D free boundary Euler equation. Evolution Equations and Control Theory, 2012, 1 (2) : 297-314. doi: 10.3934/eect.2012.1.297

[19]

Micah Webster, Patrick Guidotti. Boundary dynamics of a two-dimensional diffusive free boundary problem. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 713-736. doi: 10.3934/dcds.2010.26.713

[20]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2739-2776. doi: 10.3934/dcds.2020384

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (88)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]