Acta Mathematica Hungarica

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

On nonlinear parabolic variational inequalities containing nonlocal terms

  • Á. Besenyei


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:
$$ \begin{array}{*{20}c} {D_t u(t,x) - \sum\limits_{i = 1}^n {D_i \left[ {a_i (t,x,u(t,x),Du(t,x);u)} \right]} } \\ { + a_0 (t,x,u(t,x),Du(t,x);u) = f(t,x)} \\ {(t,x) \in Q_T = (0,T) \times \Omega ,a_i :Q_T \times R^{n + 1} \times L^p (0,T;V) \to R,} \\ \end{array} $$
where V denotes a closed linear subspace of the Sobolev-space W 1,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 


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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

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