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Long time behaviors for 3D cubic damped Klein-Gordon equations in inhomogeneous mediums

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.