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

S. A. Vakulenko

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 11, Tome 104 (1981), pp. 84-92

Résumé

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.