Justification of asymptotic formula for the solutions of perturbed Fock–Klein–Gordon equation

The Fock–Klein–Gordon equation, perturbed by the small non-linear operator $\varepsilon R[\varepsilon t,u,u_x,u_{xx}]$ is considered:
$$ u_{tt}-c^2u_{xx}+m^2u=\varepsilon R[\varepsilon t,u,u_x,u_{xx}],\quad0<\varepsilon\ll1. $$
The boundary condition and the initial data are periodical
$$ u(x+2\pi)=u(x),\quad u\mid_{t=0}a\cos x,\quad u_t\mid_{t=0}=a\omega\sin x,\quad\omega^2=c^2+m^2. $$
It is proved (if some additional conditions are realised) that 1) the solution of the problem exists on an interval $0\le t\le\ell/\varepsilon$, $\ell=\operatorname{const}>0$ and that 2) the difftrence between $u$ and the known asymptotic solution of the problem is small.