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

Abstract

Abstract. The Cauchy problem for the damped Boussinesq equation 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.

Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation

“…There is a considerable mathematical interest in the Boussinesq equations which have been studied from various aspects (see [7,8,21,22] and references therein). A great deal of efforts has been made to establish sufficient conditions for the nonexistence of global solutions to various associated boundary value problems [10,21].…”

“…Varlamov [22,23] investigated the long-time behavior of solutions to initial value, spatially periodic, and initial-boundary value problems for Eq. (1.8) in two space dimensions.…”

Global Existence and Asymptotic Behavior of Solutions to the Generalized Damped Boussinesq Equation

“…Taking into account dispersion and nonlinearity, but in real processes viscosity also plays an important role. Varlamov considered the following damped Boussinesq equation (see [2][3][4]):…”

“…Existence and uniqueness of the classical solution for the problem (4), (2) in two space dimensions was proved, and the solution was 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 was given (see [4]).…”

Asymptotic behaviors of solutions to a sixth-order Boussinesq equation with logarithmic nonlinearity

“…For this reason, Xu [38] considered the case f (u) = ±|u| p or f (u) = ±|u| p−1 u, and showed the existence of global solutions and finite time blowup solutions with suitable assumptions on the initial data. The Cauchy problem of (1.6) with n ≥ 2, a = c = d = 0 was studied in [17,25,29,35], where, with small initial data, the global well-posedness and asymptotic behavior of solutions were obtained by the semigroup method, the Green function method and the energy estimate. The Cauchy problem of (1.6) in R and R n with c = d = 0 was studied in [33] and [22], respectively.…”