Advanced Search
Article Contents
Article Contents

Parabolic quasi-variational diffusion problems with gradient constraints

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: 35K35, 35K55, 34K40, 35K87.


    \begin{equation} \\ \end{equation}
  • [1]

    A. Azevedo and L. Santos, A diffusion problem with gradient constraint depending on the temperature, Adv. Math. Sci. Appl., 20 (2010), 151-166.


    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), 107-137.doi: 10.1016/0022-1236(79)90028-4.


    A. Kadoya, Y. Murase and N. Kenmochi, A class of nonlinear parabolic systems with environmental constraints, Adv. Math. Sci. Appl., 20 (2010), 281-313.


    R. Kano, Applications of abstract parabolic quasi-variational inequalities to obstacle problems, in "Nonlocal and Abstract Parabolic Equations and their Applications," Banach Center Publ., 86 (2009), 163-174.


    R. Kano, N. Kenmochi and Y. Murase, Existence theorems for elliptic quasi-variational inequalities in Banach spaces, in "Recent Advances in Nonlinear Analysis," World Scientific, (2008) 149-169.


    R. Kano, Y. Murase and N. Kenmochi, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints, in "Nonlocal and Abstract Parabolic Equations and their Applications," Banach Center Publ., 86 (2009), 175-194.


    N. Kenmochi, Some nonlinear parabolic variational inequalities, Israel J. Math., 22 (1975), 304-331.doi: 10.1007/BF02761596.


    N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Faculty of Education, Chiba Univ., 30 (1981), 1-87.


    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), 897-908.doi: 10.1002/1099-1476(20000710)23:10<897::AID-MMA141>3.0.CO;2-H.


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


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


    Y. Murase, Abstract quasi-variational inequalities of elliptic type and applications, in "Nonlocal and Abstract Parabolic Equations and their Applications," Banach Center Publ., 86 (2009), 235-246.


    L. Prigozhin, On the Bean critical-state model in superconductivity, European J. Appl. Math., 7 (1996), 237-247.


    J.-F. Rodrigues and L. Santos, A parabolic quasi-variational inequality arising in a superconductivity model, Ann. Scuola Norm. Sup. Pisa, 29 (2000), 153-169.


    L. Santos, A diffusion problem with gradient constraint and evolutive Dirichlet condition, Portugal Math., 48 (1991), 441-468.


    L. Santos, Variational problems with non-constant gradient constraints, Portugal Math., 59 (2002), 205-248.

  • 加载中

Article Metrics

HTML views() PDF downloads(99) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint