Acta Mathematica Hungarica

, Volume 116, Issue 1–2, pp 145–162 | Cite as

On nonlinear parabolic variational inequalities containing nonlocal terms

  • Á. Besenyei
Article

Abstract

We investigate nonlinear parabolic variational inequalities which contain functional dependence on the unknown function. Such parabolic functional differential equations were studied e.g. by L. Simon in [8] (which was motivated by the work of M. Chipot and L. Molinet in [4]), where the following equation was considered:
$$ \begin{array}{*{20}c} {D_t u(t,x) - \sum\limits_{i = 1}^n {D_i \left[ {a_i (t,x,u(t,x),Du(t,x);u)} \right]} } \\ { + a_0 (t,x,u(t,x),Du(t,x);u) = f(t,x)} \\ {(t,x) \in Q_T = (0,T) \times \Omega ,a_i :Q_T \times R^{n + 1} \times L^p (0,T;V) \to R,} \\ \end{array} $$
(1)
where V denotes a closed linear subspace of the Sobolev-space W1,p(Ω) (2 ≦ p < ∞). In the above mentioned paper existence of weak solutions of the above equation is shown. These results were extended to systems of functional differential equations in [2]. In the following, we extend these existence results to variational inequalities by using the (less known) results of [6]. Finally, we show some examples.

Key words and phrases

parabolic variational inequalities nonlocal evolution problem monotone operators 

2000 Mathematics Subject Classification

35K85 49J40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. Berkovits and V. Mustonen, Topological degree for perturbations of linear maximal monotone mappings and applications to a class of parabolic problems, Rend. Mat. Ser. VII, Roma, 12 (1992), 597–621.MATHMathSciNetGoogle Scholar
  2. [2]
    A. Besenyei, On systems of parabolic functional differential equations, Annales Univ. Sci. Budapest., 47 (2004), 143–160.MathSciNetGoogle Scholar
  3. [3]
    F. E. Browder, Pseudo-monotone operators and nonlinear elliptic boundary value problems on unbounded domains, Proc. Natl. Acad. Sci. USA, 74 (1977), 2659–2661.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    M. Chipot and L. Molinet, Asymptotic behavior of some nonlocal diffusion problems, Applicable Analysis, 80 (2001), 279–315.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    J. L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, Gauthier-Villars (Paris, 1969).Google Scholar
  6. [6]
    J. Naumann, Einführung in die Theorie parabolischer Variationsungleichungen, Teubner-Texte zur Mathematik, 64 (Leipzig, 1984).Google Scholar
  7. [7]
    D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Edit. Academiei Bucaresti, Sijthoff & Noordhoff Intern. Publ. (1978).Google Scholar
  8. [8]
    L. Simon, On parabolic functional differential equations of general divergence form, in: Proceedings of the Conference Function Spaces, Differential Operators and Nonlinear Analysis (Milovy, 2004), pp. 280–291.Google Scholar
  9. [9]
    E. Zeidler, Nonlinear Functional Analysis and its Applications II–III, Springer (1990).Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  • Á. Besenyei
    • 1
  1. 1.Institute of Mathemaics, Department of Applied AnalysisEötvös Loránd UniversityBudapestHungary

Personalised recommendations