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On the stability of stationary solutions of first order evolution variational inequalities. (English) Zbl 1085.35090

Sufficient conditions for the Lyapunov stability, asymptotic stability and instability of the trivial solution are given for a general class of first order evolution variational inequalities in a Hilbert space for which the initial value problem has a unique strong global solution. Unilateral problems with nonlinear operators and constraints described by closed convex sets are covered. The results are based on a method of Lyapunov type function developed here for variational inequalities. For the proofs, techniques of nonlinear evolution equations are combined with variational inequalities methods.

Abstract results are applied, in particular, to variational inequalities with nonlinear operators of monotone type. The case of a finite dimensional space and linear operators (matrices) is also discussed in details. It is shown that the stability conditions for variational inequalities given by P. Quittner follow from the abstract results obtained.

MSC:
35K85Linear parabolic unilateral problems; linear parabolic variational inequalities
34D20Stability of ODE
37B25Lyapunov functions and stability; attractors, repellers
35B35Stability of solutions of PDE