# On nonlinear parabolic variational inequalities containing nonlocal terms

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## 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: where

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

(1)

*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## Preview

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## References

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