The Existence And Fractal Dimension Of The Global Attractor For A Viscoelastic Wave Equation With Nonlinear Damping

Let be a bounded domain inR3with smooth boundary. We consider the followingsemilinear wave equations with nonlinear damping and linear memory term:It is well known that nonlinear dissipation in hyperbolic and hyperbolic-like equationshas been a source of technical difficulties in the analysis of longtime dynamics of solutions.This paper is divided into two parts. In the first part, we prove the existence of the globalattractor for the equation above on the energy space of proof of the existence of the global attractor for the equation above, the first step is to showthe existence of an attracting set on the energy space, split the solution to equation into thesum: u (t)(t)(t)and prove the compactness of tand the uniform decay of t. Fromthat we get the existence of global attractor. In the second part, we estimate the fractaldimension of the global attractor. First, we introduce the concept of l-trajectories spacesE andA. Second, we give the connection between the fractal dimensions of the l-trajectoriesspaces and. Then, prove the generalized squeezing property (GSP), and hence we obtainthe finiteness of the fractal dimension.