Asymptotics of solutions with a compactness property for the nonlinear damped Klein–Gordon equation

We consider the nonlinear damped Klein–Gordon equation ∂ttu+2α∂tu−Δu+u−|u|p−1u=0on[0,∞)×RNwith α>0, 2⩽N⩽5 and energy subcritical exponents p>2. We study the behavior of solutions for which it is supposed that only one nonlinear object appears asymptotically for large times, at least for a sequence of times.