For the damped Boussinesq equation $u_{tt}-2bu_{txx}= -\alpha u_{xxxx}+ u_{xx}+\beta(u^2)_{xx},x\in(0,\pi),t > 0;\alpha,b = const > 0,\beta = const\in R^1$ , the second initial-boundary value problem is considered with small initial data. Its classical solution is constructed in the form of a series in small parameter present in the initial conditions and the uniqueness of solutions is proved. The long-time asymptotics is obtained in the explicit form and the question of the blow up of the solution in a certain case is examined. The possibility of passing to the limit $b\rightarrow + 0$ in the constructed solution is investigated.