Convergence Rate of Solutions to a Hyperbolic Equation with $p(x)$-Laplacian Operator and Non-Autonomous Damping

This paper concerns the convergence rate of solutions to a hyperbolicequation with $p(x)$-Laplacian operator and non-autonomous damping. We apply the Faedo-Galerkin method to establish the existence of global solutions,and then use some ideas from the study of second order dynamical system toget the strong convergence relationship between the global solutions and thesteady solution. Some differential inequality arguments and a new Lyapunovfunctional are proved to show the explicit convergence rate of the trajectories.