On spatially periodic solutions of the damped Boussinesq equation

Abstract

A classical solution of the damped Boussinesq equation $$ u_{tt}-2bu_{txx}=-\alpha u_{xxxx}+u_{xx}+\beta (u^2)_{xx},\quad x\in {\Bbb R}^1,t>0, $$ with $\alpha ,b=\text{const}>0$, $\beta =\text{const}\in{\Bbb R}^1$, $\alpha >b^2$, and small initial data is constructed by means of the successive application of the spectral theory and the perturbation one. Its long-time asymptotic representation is obtained which shows that the major term increases linearly with time and the second term contains a combination of the Airy functions of a negative argument. A uniform-in-space estimate of the remainder is given.