Trajectory and global attractors of dissipative hyperbolic equations with memory

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