Asymptotic expansion of solutions to the periodic problem for a non-linear Sobolev-type equation

We study the long-time behaviour of solutions to the periodic problem for a non-linear Sobolev-type equation. In the case of non-small initial perturbations we get estimates for the decay as a function of time. In the case of small initial data we prove an asymptotic formula for solutions to the periodic problem for a non-linear Sobolev-type equation.