Decay of solutions of a second order differential equation with non-smooth second member

Abstract

Consider the following abstract initial value problem(⁎)|u″(t)+μ(t)Au(t)+a(|A−θ2u(t)|2)u(t)+b(|A−ηu′(t)|)Au′(t)=f(t)in (0,∞);u(0)=u0,u′(0)=u1 in a real separable Hilbert space H with norm |u|. Here A is a positive self-adjoint operator of H; μ(t),a(s),b(s) positive functions, f(t) a vectorial non-smooth function and θ, η real numbers. In this paper we study the existence, uniqueness and decay of solutions of problem (⁎). In our approach, we use the Theory of Self-Adjoint Operators in Hilbert spaces, the compactness Aubin–Lions Theorem and a Lyapunov functional.