The global solution of an initial boundary value problem for the damped Boussinesq equation

This paper deals with an initial-boundary value problem for the pp damped Boussinesq equation u(tt) - au(ttxx) - 2bu(txx) -cu(xxxx) + u(xx) + beta(u(2))(xx), where t > 0, a, b, c and beta are constants. For the case a greater than or equal to 1 and a + c > b(2), corresponding to an infinite number of damped oscillations, we derived the global solution of the equation in the form of a Fourier series. The coefficients of the series are related to a small parameter present in the initial conditions and are expressed as uniformly convergent series of the parameter. Also we prove that the long time asymptotics of the solution in question decays exponentially in time.