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

was derived by Boussinesq [1] in 1872 to describe shallow water waves, where u(x, t) is an elevation of the free surface of fluid and the constant coefficients α and β 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]):

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][3][4]): where  > 0 and  > 0 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  > 0 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]). 2
The paper is organized as follows.In Section 2, we derive the solution formula of our semilinear problem.We study the decay property of the solution operators appearing in the solution formula in Section 3.Then, in Section 4, we prove the global existence and asymptotic decay of solutions.Finally, we derive a simpler asymptotic profile which gives the approximation to the linear solution in Section 5.
Notations.We give some notations which are used in this paper.Let F[] denote the Fourier transform of  defined by and we denote its inverse transform by F −1 .
Finally, in this paper, we denote every positive constant by the same symbol  or  without confusion.[⋅] is the Gauss symbol.

Solution Formula
The aim of this section is to derive the solution formula for the problem (1), (2).We first investigate the linearized equation of (1): with the initial data in (2).We apply the Fourier transform to (8).This yields The corresponding initial value is given as The characteristic equation of ( 9) is Let  =  ± () be the corresponding eigenvalues of (11), and we obtain The solution to the problem ( 9), (10) in the Fourier space is then given explicitly in the form û (, ) = Ĝ (, ) û1 () + Ĥ (, ) û0 () , where We define (, ) and (, ) by respectively, where F −1 denotes the inverse Fourier transform.Then, applying F −1 to (13), we obtain By the Duhamel principle, we obtain the solution formula to (1), (2) as follows:

Decay Property
The aim of this section is to establish decay estimates of the solution operators () and () appearing in (16)  for  ∈ R  and  ≥ 0.
Proof.Multiplying ( 9) by û and taking the real part yield Multiplying ( 9) by û and taking the real part, we obtain Multiplying both sides of ( 22) by || 2 and summing up the resulting equation and ( 21) yield where A simple computation implies that where Note that It follows from (25) that Using ( 23) and (28), we get Thus which together with (25) proves the desired estimates (20).
Then we have completed the proof of the lemma.
Lemma 2. Let Ĝ(, ) and Ĥ(, ) be the fundamental solution of (8) in the Fourier space, which are given in (14) and (15), respectively.Then one has the estimates for  ∈ R  and  ≥ 0.
Combining the previous three inequalities yields (35).This completes the proof of Lemma 3.
From Lemma 3, we immediately have the following corollary.

Global Existence and Asymptotic Behavior of Solutions to (1), (2)
The purpose of this section is to prove global existence and asymptotic behavior of solutions to the Cauchy problem (1), (2).We need the following lemma, which come from [18] (see also [19]).
The previous proof of Theorem 6 shows that when  ≥ 2, the solution  to the integral equation ( 19) is asymptotic to the linear solution   given by the formula   () := () *  1 + () *  0 in (18) as  → ∞.This result is stated as follows.

Asymptotic Linear Profile
In the previous section, we have shown that the solution  to the problem (1), (2) can be approximated by the linear solution   .The aim of this section is to derive a simpler asymptotic profile of the linear solution   .