Trajectory and global attractors of dissipative hyperbolic equations with memory

We consider in this article a general construction of trajectory attractorsand global attractors of evolution equations with memory. In our approach,the corresponding dynamical system acts in the space of initial data of theCauchy problem under study; we can note that, in previous studies, theso-called history space setting was introduced and the study of globalattractors was made in an extended phase space.As an application, we construct trajectory and global attractors fordissipative hyperbolic equations with linear memory. We also prove theexistence of a global Lyapunov function for the dissipative hyperbolicequation with memory. The existence of such a Lyapunov function implies aregular structure for the trajectory and global attractors of the equationunder consideration.