Asymptotic Decomposition for Nonlinear Damped Klein-Gordon Equations (Journal of Mathematical Study, 2020)
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 isthe existence of the "concentration-compact attractor” introduced by Tao which yieldsa finite number of asymptotic profiles. Using the damping effect, we can prove all theprofiles are equilibrium points.
Keywords: Nonlinear Klein-Gordon equations; damping; soliton resolution; global attractor.