Asymptotic behavior for a nondissipative and nonlinear system of the Kirchhoff viscoelastic type

A wave equation of the kirchhoff type with several nonlinearities is stabilized by a viscoelastic damping. we consider the case of nonconstant (and unbounded) coefficients. this is a nondissipative case,and as a consequence the nonlinear terms cannot be estimated in the usual manner by the initial energy. we suggest a way to get around this difficulty. it is proved that if the solution enters a certain region,which we determine,then it will be attracted exponentially by the equilibrium. © 2012 nasser-eddine tatar.