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.
Nasser-Eddine Tatar. "Asymptotic Behavior for a Nondissipative and Nonlinear System of the Kirchhoff Viscoelastic Type." J. Appl. Math. 2012 1 - 17, 2012. https://doi.org/10.1155/2012/936140