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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 & Heterogeneous Media, 2014, 9 (4) : 683-707. doi: 10.3934/nhm.2014.9.683
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References:
 [1] H. Gajewski, I. V. Skrypnik. To the uniqueness problem for nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 315-336. doi: 10.3934/dcds.2004.10.315 [2] Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1081-1104. doi: 10.3934/dcds.2003.9.1081 [3] Peiying Chen. Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations. Electronic Research Announcements, 2017, 24: 38-52. doi: 10.3934/era.2017.24.005 [4] Yucheng Bu, Yujun Dong. Existence of solutions for nonlinear operator equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4429-4441. doi: 10.3934/dcds.2019180 [5] Stanislav Antontsev, Michel Chipot, Sergey Shmarev. Uniqueness and comparison theorems for solutions of doubly nonlinear parabolic equations with nonstandard growth conditions. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1527-1546. doi: 10.3934/cpaa.2013.12.1527 [6] Filippo Gazzola. On the moments of solutions to linear parabolic equations involving the biharmonic operator. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3583-3597. doi: 10.3934/dcds.2013.33.3583 [7] Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781 [8] Peter Poláčik. On uniqueness of positive entire solutions and other properties of linear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 13-26. doi: 10.3934/dcds.2005.12.13 [9] Juncheng Wei, Wei Yao. Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1003-1011. doi: 10.3934/cpaa.2012.11.1003 [10] Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771 [11] Shouchuan Hu, Nikolaos S. Papageorgiou. Nonlinear Neumann equations driven by a nonhomogeneous differential operator. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1055-1078. doi: 10.3934/cpaa.2011.10.1055 [12] Fabio Punzo. Support properties of solutions to nonlinear parabolic equations with variable density in the hyperbolic space. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 657-670. doi: 10.3934/dcdss.2012.5.657 [13] H. Merdan, G. Caginalp. Decay of solutions to nonlinear parabolic equations: renormalization and rigorous results. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 565-588. doi: 10.3934/dcdsb.2003.3.565 [14] Huijiang Zhao. Large time decay estimates of solutions of nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 69-114. doi: 10.3934/dcds.2002.8.69 [15] Mitsuharu Ôtani, Yoshie Sugiyama. Lipschitz continuous solutions of some doubly nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 647-670. doi: 10.3934/dcds.2002.8.647 [16] P. R. Zingano. Asymptotic behavior of the $L^1$ norm of solutions to nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 151-159. doi: 10.3934/cpaa.2004.3.151 [17] Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395 [18] Pavel Gurevich. Periodic solutions of parabolic problems with hysteresis on the boundary. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1041-1083. doi: 10.3934/dcds.2011.29.1041 [19] N. V. Krylov. Uniqueness for Lp-viscosity solutions for uniformly parabolic Isaacs equations with measurable lower order terms. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2495-2516. doi: 10.3934/cpaa.2018119 [20] Jan Prüss, Gieri Simonett, Rico Zacher. On normal stability for nonlinear parabolic equations. Conference Publications, 2009, 2009 (Special) : 612-621. doi: 10.3934/proc.2009.2009.612

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