Compactness of trajectories to some nonlinear second order evolution equations and applications

Abstract

Under suitable growth and coercivity conditions on the nonlinear damping operator g, we establish boundedness or compactness properties of trajectories to the equationu¨(t)+g(u˙(t))+Au(t)=h(t),t∈R+, where A is a positive selfadjoint operator. The compactness results are used to prove the existence of almost periodic solutions when h is almost periodic, and to generalize some recent results of Chergui and Ben Hassen–Chergui concerning convergence to equilibrium when a nonlinear term depending on u is added and h dies off sufficiently fast for t large.