Existence and Decay of Solutions of Some Nonlinear Parabolic Variational Inequalities

This paper discusses the existence and decay of solutions u(t) of the variational inequality of parabolic type: <u^’(t) + Au(t) + Bu(t) − f(t), v(t) − u(t)> ≧ 0 for ∀ v ∈ L^P([0,∞);V (p≧2) with v(t) ∈ K a.e. in [0,∞), where K is a closed convex set of a separable uniformly convex Banach space V, A is a nonlinear monotone operator from V to V* and B is a nonlinear operator from Banach space W to W*. V and W are related as V ⊂ W ⊂ H for a Hilbert space H. No monotonicity assumption is made on B.