Asymptotic behavior of solutions of the damped Boussinesq equation in two space dimensions.

*(English)*Zbl 0919.35111Summary: The Cauchy problem for the damped Boussinesq equation

$${u}_{tt}=-\alpha {u}_{xxxx}+{u}_{xx}+\beta {\left({u}^{2}\right)}_{xx},\phantom{\rule{1.em}{0ex}}x\in {\mathbb{R}}^{1},\phantom{\rule{1.em}{0ex}}t>0,$$

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.