Long-time Asymptotic for the Damped Boussinesq Equation in a Circle

Abstract

The first initial-boundary value problem for the following equation u_{tt} - aΔu_{tt} - 2bΔu_t = αΔ^3u - βΔ²u + Δu + ϒΔ(u²) in a unit circle is considered. The existence of strong solution is established in the space C^0([0, ∞), H^s_r (0, 1)), s < 7/2, and the solutions are constructed in the form of series in the small parameter present in the initial conditions. For 5/2 < s < 7/2, the uniqueness is proved. The long-time asymptotics is obtained in the explicit form.