Long-time asymptotics of solutions of the second initial-boundary value problem for the damped Boussinesq equation

Abstract

For the damped Boussinesq equation $u_{tt}-2b{u}_{txx}=-\alpha u_{xxxx}+u_{xx}+\beta {\left(u^2\right)}_{xx},x\in \left(0,\pi \right),t>0;\alpha ,b=const>0,\beta =const\in R^1$, 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 $b\to +0$ in the constructed solution is investigated.