Parabolic variational inequalities with generalized reflecting directions

We study, in a Hilbert framework, some abstract parabolic variational inequalities, governed by reflecting subgradients with multiplicative perturbation, of the following type: y'(t) + Ay(t) + Theta(t, y, (t))partial derivative phi(y(t)) f(t, y(t)), y(0) = y(0), t is an element of[0, T], where A is a linear self-adjoint operator, partial derivative phi is the subdifferential operator of a proper lower semicontinuous convex function phi defined on a suitable Hilbert space, and Theta is the perturbing term which acts on the set of reflecting directions, destroying the maximal monotony of the multivalued term. We provide the existence of a solution for the above Cauchy problem. Our evolution equation is accompanied by examples which aim to (systems of) PDEs with perturbed reflection.