Asymptotic Decomposition for Nonlinear Damped Klein-Gordon Equations

In this paper, we prove that if the solution to the damped focusing Klein-Gordon equations is global forward in time with bounded trajectory, then it will decouple into the superposition of divergent equilibriums. The core ingredient of our proof is the existence of the "concentration-compact attractor" introduced by Tao which yields a finite number of asymptotic profiles. Using the damping effect, we can prove all the profiles are equilibrium points.