Existence of asymptotically almost periodic solutions for damped wave equations

In this paper, a class of nonlinear damped wave equations of the form α u ‴ ( t ) + u ″ ( t ) = β A u ( t ) + γ A u ′ ( t ) + f ( t , u ( t ) ) , t ⩾ 0 , satisfying α β < γ with prescribed initial conditions are studied. Some sufficient conditions are established for the existence and uniqueness of an asymptotically almost periodic solution. These results have significance in the study of vibrations of flexible structures possessing internal material damping. Finally, an example is presented to illustrate the feasibility and effectiveness of the results.