Global Existence and Decay Rates of Solutions of Generalized Benjamin–Bona–Mahony Equations in Multiple Dimensions

We study the global existence and decay rates of the Cauchy problem for the generalized Benjamin–Bona–Mahony equations in multi-dimensional spaces. By using Fourier analysis, frequency decomposition, pseudo-differential operators and the energy method, we obtain global existence and optimal L2 convergence rates of the solution.


Introduction
In this paper, we study the global existence and decay rates of the smooth solution u(x, t) to the scalar multi-dimensional generalized Benjamin-Bona-Mahony (GBBM) equations of the form ∂ t u − ∂ t u − η u + (β · ∇)u + div f (u) = 0, u(0, x) = u 0 (x). (1.1) Here η is a positive constant, β is a real constant vector, f (u) = (u 2 , u 2 , . . . , u 2 ), = n j =1 ∂ 2 x j is the Laplacian, ∇ is a gradient operator, n ≥ 2 is the spatial dimension. The well known Benjamin-Bona-Mahony (BBM) equation and its counterpart, the Korteweg-de Vries (KdV) equation were both suggested as model equations for long waves in nonlinear dispersive media. The BBM equation was advocated by Benjamin, Bona and Mahony [3] in 1972. Since then, the periodic boundary value problems, the initial value problems and the initial boundary value problems, for various generalized BBM equations have been studied. The existence and uniqueness of solutions for GBBM have been proved by many authors [2,3,7,8]. The decays of solutions were also studied in [1,[4][5][6]10]. In [1,4,5,10], the authors studied the equation in low spatial dimension. In [6], the equation in high spatial dimension was studied, but the authors just got a global solution u(x, t) ∈ H 1 . In this paper, the goal is to get a global smoother solution and give its L 2 decay rates in high spatial dimension. Throughout this paper, we denote the generic constants by C and write D α f = ∂ α 1 In particular, W s,2 = H s . The Fourier transformation with respect to the variable x ∈ R n iŝ and the inverse Fourier transformation with respect to the variable ξ is In this paper, all convolutions are only with respect to the spatial variable x.
Our main result is the following: Remark The decay rate is the same as that of the heat equation, so our estimate is optimal.
The rest of the paper is arranged as follows. In Sect. 2, we get the local existence of the solution directly by constructing a Cauchy sequence and using energy estimation. In Sect. 3, by means of the Green function, we obtain a bound of the solution, then we extend the local solution to the global one. In Sect. 4, we get an L 2 decay estimate of the solution.

Local existence
In this section, we will construct a convergent sequence {u (m) (x, t)} to get the local solution, where u (m) (x, t) satisfy the following linear problem (2.1) We will construct a Banach space and prove that the sequence is convergent in this space, so the limit is the solution of (1.1). First, we define a function space and n ≥ 2. The metric in X T 0 is induced by the norm u X T 0 : It is obvious that X T 0 is a nonempty complete space.
Proof We will prove this lemma by induction on m. When m = 1, we have Multiplying the first equation by u (1) and integrating with respect to the variable x in R n , we get here we use the fact that (β · ∇u (1) ) · u (1) dx = 0. Then Multiplying the two sides in (2.3) by D α x u (1) and integrating with respect to the variable x in R n , we also get Thus, for every T 1 , we have u (1) ∈ X T 1 . Supposing that u (m) (x, t) ∈ X T 1 , we next prove that u (m+1) (x, t) ∈ X T 1 . By definition, u (m+1) (x, t) satisfies the following equation: To proceed, we need the following inequality: where 1 ≤ r, p, q ≤ ∞ and 1 (2.7) By the Sobolev embedding inequality, we have By the induction method, our lemma is proved. Proof We only need to prove that for some 0 < λ < 1. By (2.1), for every m, u (m+1) − u (m) satisfies the following equation: (2.10) Thus we have Multiplying the two sides of (2.11) by D α x (u (m+1) − u (m) ) and integrating with respect to the variable x in R n , we get Noticing that (u (m+1) − u (m) )(0, x) = 0, and choosing T 0 = min(T 1 , 1 4CE 2 ), we have Thus we can get (2.9).
Since X T 0 is a complete metric space, by Lemmas 2.1 and 2.2, there exists a u(x, t) ∈ X T 0 such that Thus we have proved the local existence of (1.1). Next we will prove the global existence.

Bounded estimates and global existence
In this section, we want to get u H l ≤ CE, then extend the local solution to a global one.
Proof Multiplying the first equation by u and integrating with respect to the variable x in R n , we get Thus and our lemma is proved.
In order to get a bound of u, first we want to get an explicit expression of u through the Green function. The linearized system of (1.1) is of the form The Green function of (3.2) is defined as Direct calculation shows that the Fourier transformation of G iŝ Due to Duhamel's principle, we know that the solution of (1.1) can be expressed by Next we want to analyze u through estimates of G, H . For this part, we must estimate decay rates ofĜ,Ĥ separately for low and high frequencies. Thus we set where χ 1 , χ 2 are smooth cut-off functions and χ 1 (ξ ) + χ 2 (ξ ) = 1. SetĜ i = χ iĜ ,Ĥ i = χ iĤ . For low frequencies, we have the following lemma.

Lemma 3.2 For
Proof By the Hausdorff-Young inequality, if the integers p ≥ 2 and q satisfy 1 p + 1 q = 1, then we have Next we analyze the constructions of G, H for high frequencies. ). Here Proof We just prove the first inequality; the proofs of the others are similar. By (3.3), if |β| ≥ 1 then we have From this and Lemma 3.2 in [9], we get our result.
According to Lemma 3.5 and using the local solution, by the usual method, we can derive a global solution for (1.1) such that u ∈ L ∞ (0, ∞, H l (R n )).

Decay estimation
Setting M(t) = sup |α|≤l,0≤s≤t D α x u L 2 (1 + s) Thus we get Theorem 1.1, our main result in this paper.