Asymptotic behavior of solutions of the damped Boussinesq equation in two space dimensions (Q1283413)

Summary: The Cauchy problem for the damped Boussinesq equation \[ u_{tt}=-\alpha u_{xxxx}+ u_{xx}+ \beta(u^2)_{xx},\quad x\in\mathbb{R}^1,\quad t>0, \] with small initial data is considered in two space dimensions. Existence and uniqueness of its classical solution is proved and the solution is constructed in the form of a series. The major term of its long-time asymptotics is calculated explicitly and a uniform in space estimate of the residual term is given.