April  2013, 6(2): 423-438. doi: 10.3934/dcdss.2013.6.423

Parabolic quasi-variational diffusion problems with gradient constraints

1. 

Department of Education, School of Education, Bukkyo University, 96 Kitahananobo-cho, Murasakino, Kita-ku, Kyoto, 603-8301

Received  July 2011 Revised  December 2011 Published  November 2012

In this paper we consider some mechanical phenomena whose dynamics is described by a class of quasi-variational inequalities of parabolic type. Our system consists of a second-order parabolic variational inequality with gradient constraint depending on the temperature and the heat equation. Since the temperature is unknown in our problem, the constraint function is unknown as well. In this sense, our problem includes the quasi-variational structure, and in the mathamtical analysis one of main difficulties comes from it. Our approach to the problem is based on the abstract theory of quasi-variational inequalities with non-local constraint which has been developed in [6]. However the abstract theory is not directly used in the existence proof of a solution, since the mathematical situation of the problem is much nicer than that in the abstract theory [6]. In this paper we prove the existence of a weak solution of our system.
Citation: Nobuyuki Kenmochi. Parabolic quasi-variational diffusion problems with gradient constraints. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 423-438. doi: 10.3934/dcdss.2013.6.423
References:
[1]

A. Azevedo and L. Santos, A diffusion problem with gradient constraint depending on the temperature,, Adv. Math. Sci. Appl., 20 (2010), 151. Google Scholar

[2]

J. L. Joly and U. Mosco, A propos de l'existence et de la régularité des solutions de certaines inéquations quasi-variationnelles,, J. Funct. Anal., 34 (1974), 107. doi: 10.1016/0022-1236(79)90028-4. Google Scholar

[3]

A. Kadoya, Y. Murase and N. Kenmochi, A class of nonlinear parabolic systems with environmental constraints,, Adv. Math. Sci. Appl., 20 (2010), 281. Google Scholar

[4]

R. Kano, Applications of abstract parabolic quasi-variational inequalities to obstacle problems,, in, 86 (2009), 163. Google Scholar

[5]

R. Kano, N. Kenmochi and Y. Murase, Existence theorems for elliptic quasi-variational inequalities in Banach spaces,, in, (2008), 149. Google Scholar

[6]

R. Kano, Y. Murase and N. Kenmochi, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints,, in, 86 (2009), 175. Google Scholar

[7]

N. Kenmochi, Some nonlinear parabolic variational inequalities,, Israel J. Math., 22 (1975), 304. doi: 10.1007/BF02761596. Google Scholar

[8]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications,, Bull. Faculty of Education, 30 (1981), 1. Google Scholar

[9]

M. Kunze and J.-F. Rodrigues, An elliptic quasi-variational inequality with gradient constraints and some of its applications,, Math. Mech. Appl. Sci., 23 (2000), 897. doi: 10.1002/1099-1476(20000710)23:10<897::AID-MMA141>3.0.CO;2-H. Google Scholar

[10]

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

[11]

J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications, Volume I,", Springer, (1972). Google Scholar

[12]

Y. Murase, Abstract quasi-variational inequalities of elliptic type and applications,, in, 86 (2009), 235. Google Scholar

[13]

L. Prigozhin, On the Bean critical-state model in superconductivity,, European J. Appl. Math., 7 (1996), 237. Google Scholar

[14]

J.-F. Rodrigues and L. Santos, A parabolic quasi-variational inequality arising in a superconductivity model,, Ann. Scuola Norm. Sup. Pisa, 29 (2000), 153. Google Scholar

[15]

L. Santos, A diffusion problem with gradient constraint and evolutive Dirichlet condition,, Portugal Math., 48 (1991), 441. Google Scholar

[16]

L. Santos, Variational problems with non-constant gradient constraints,, Portugal Math., 59 (2002), 205. Google Scholar

show all references

References:
[1]

A. Azevedo and L. Santos, A diffusion problem with gradient constraint depending on the temperature,, Adv. Math. Sci. Appl., 20 (2010), 151. Google Scholar

[2]

J. L. Joly and U. Mosco, A propos de l'existence et de la régularité des solutions de certaines inéquations quasi-variationnelles,, J. Funct. Anal., 34 (1974), 107. doi: 10.1016/0022-1236(79)90028-4. Google Scholar

[3]

A. Kadoya, Y. Murase and N. Kenmochi, A class of nonlinear parabolic systems with environmental constraints,, Adv. Math. Sci. Appl., 20 (2010), 281. Google Scholar

[4]

R. Kano, Applications of abstract parabolic quasi-variational inequalities to obstacle problems,, in, 86 (2009), 163. Google Scholar

[5]

R. Kano, N. Kenmochi and Y. Murase, Existence theorems for elliptic quasi-variational inequalities in Banach spaces,, in, (2008), 149. Google Scholar

[6]

R. Kano, Y. Murase and N. Kenmochi, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints,, in, 86 (2009), 175. Google Scholar

[7]

N. Kenmochi, Some nonlinear parabolic variational inequalities,, Israel J. Math., 22 (1975), 304. doi: 10.1007/BF02761596. Google Scholar

[8]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications,, Bull. Faculty of Education, 30 (1981), 1. Google Scholar

[9]

M. Kunze and J.-F. Rodrigues, An elliptic quasi-variational inequality with gradient constraints and some of its applications,, Math. Mech. Appl. Sci., 23 (2000), 897. doi: 10.1002/1099-1476(20000710)23:10<897::AID-MMA141>3.0.CO;2-H. Google Scholar

[10]

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

[11]

J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications, Volume I,", Springer, (1972). Google Scholar

[12]

Y. Murase, Abstract quasi-variational inequalities of elliptic type and applications,, in, 86 (2009), 235. Google Scholar

[13]

L. Prigozhin, On the Bean critical-state model in superconductivity,, European J. Appl. Math., 7 (1996), 237. Google Scholar

[14]

J.-F. Rodrigues and L. Santos, A parabolic quasi-variational inequality arising in a superconductivity model,, Ann. Scuola Norm. Sup. Pisa, 29 (2000), 153. Google Scholar

[15]

L. Santos, A diffusion problem with gradient constraint and evolutive Dirichlet condition,, Portugal Math., 48 (1991), 441. Google Scholar

[16]

L. Santos, Variational problems with non-constant gradient constraints,, Portugal Math., 59 (2002), 205. Google Scholar

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