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

We investigate the Cauchy problem for the generalized damped Boussinesq equation. Under small condition on the initial value, we prove the global existence and optimal decay estimate of solutions for all space dimensions . Moreover, when , we show that the solution can be approximated by the linear solution as time tends to infinity. 1. Introduction We investigate the Cauchy problem of the following generalized damped Boussinesq equation: with the initial value Here is the unknown function of and , , and are constants. The nonlinear term is a given smooth function of satisfying for . It is well known that the classical Boussinesq equation was derived by Boussinesq [1] in 1872 to describe shallow water waves, where is an elevation of the free surface of fluid and the constant coefficients and depend on the depth of fluid and the characteristic speed of long waves. It is interesting to note that this equation also governs nonlinear string oscillations. Taking into account dispersion and nonlinearity, but in real processes viscosity also plays an important role. Varlamov considered the following damped Boussinesq equation (see [2–4]): where and are constants. Under small condition on the initial value, Varlamov [2] obtained a classical solution to the problem (4), (2) by means of the application of both the spectral and perturbation theories. Moreover, large time asymptotics of this solution was also discussed. For the problem (4), (2) in one, two, and three space dimensions, existence and uniqueness of local solution are proved by Varlamov [3]. The author also showed that for discontinuous initial perturbations this solution is infinitely differentiable with respect to time and space coordinates for on a bounded time interval. 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]). The main purpose of this paper is to establish the following optimal decay estimate of solutions to (1), (2) for : for and . Here is assumed to be small. Moreover, when , we show that our solution can be approximated by the solution to the linearized problem, namely, the problem (1), (2) with . More precisely, when , we show that for and , where for and for . The study of the global existence and asymptotic behavior of solutions to hyperbolic-type equations has a long history. We refer to [5, 6] for hyperbolic equations,