Decay of Solutions of Damped Kirchhoff and Beam Equations

Abstract

We obtain uniform estimates for solutions of second-order nonlinear nonautonomous differential-operator equation in a Hilbert space with structural damping. It is shown that when the given source term in the equation tends to zero as t -> infinity, the corresponding solution of the Cauchy problem for this equation also tends to zero as t -> infinity. Exponential decay of solutions for the corresponding autonomous equation is also obtained. Applications to the initial boundary value problems for some nonlinear Kirchhoff type and beam equations are given.