On the initial boundary value problem for the damped Boussinesq equation

The first initial-boundary value problem for the damped Boussinesq equation ut t-2but x x=-α ux x x x+ux x+β (u²)x x, x∈ (0,π), t>0, with α, b=const>0, β =const∈ R¹, is considered withsmall initial data. For the most interesting case α >b²corresponding to an infinite number of damped oscillations its solution isconstructed in the form of a Fourier series which coefficients in their ownturn are represented as series in small parameter present in the initialconditions. The solution of the corresponding problem for the classicalBoussinesq equation on [0,T], T<+∞, is obtained by means of passingto the limit b→ +0. Long-time asymptotics of the solution inquestion is calculated which shows the presence of the damped oscillationsdecaying exponentially in time. This is in contrast with the long timebehavior of the solution of the periodic problem studied in [30] which majorterm increases linearly with time.