  In this paper, we study the asymptotic dynamics of global solutions to damped
Klein-Gordon equations in inhomogeneous mediums (KGI). In the defocusing case,
we prove for any initial data, the solution is globally define in forward time
and it will converge to an equilibrium. In the focusing case, for global
solutions, we prove the solutions converge to the superposition of equilibriums
among which there exists at most one equilibrium to KGI and the other
equilibriums are solutions to stationary nonlinear Klein-Gordon equations. The
core ingredients of our proof are the existence of the "concentration-compact
attractor" and the gradient system theory.
