Existence and decay of solutions of some nonlinear parabolic variational inequalities

Abstract

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)>≧0for ∀v∈Lp([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.