On Time Periodic Solutions to the Generalized BBM-Burgers Equation with Time-Dependent Periodic External Force

In this paper, we consider the generalized BBM-Burgers equation with periodic external force in R. Existence and uniqueness of time periodic solutions that have the same period as the external force are established in some suitable function space for the space dimension n ≥ 3. Moreover, we also discuss the time asymptotic stability of the time periodic solution. The proof is mainly based on the contraction mapping theorem and continuous argument.


Introduction
In this paper, we investigate existence and asymptotic stability of time periodic solutions to the generalized BBM-Burgers equation with time-dependent periodic external force (1.1) When h j = 0, Geng and Chen [3] proved existence and the uniqueness of the global generalized solution and the global classical solution for the Cauchy problem of equation (1.1) for n ≤ 3. The proof is based by existence of local solutions and energy method. Moreover, the decay property of the solution was discussed. When g = h j = 0, Zhao [18] proved the existence and convergence of the global smooth solutions to (1.1). For the multidimensional case, we refer to Zhao [16] and [17]. When γ = 0 and h j = 0, Chen and Xue [2] proved that global existence and asymptotic behavior of solutions in one space dimension. Later, global existence and optimal decay estimate of solution have been established in [13]. Moreover, they also showed that as time tends to infinity, the global solution approaches the nonlinear diffusion wave described by the self-similar solution of the viscous Burgers equation for n = 1. For more study of other type Benjamin-Bona-Mahony-Burgers equations, we may refer to [1], [4], [6], [7], [8] and [15].
Is there time periodic solution to (1.1), which has the same period as h j (j = 1, . . . , n)? Is the time periodic solution unique and stable? To the best of our knowledge, these are interesting and challenging open problems and few results are available. We shall try to solve these problems in this paper. So our main purpose of this paper is to establish existence, uniqueness and asymptotic stability of time periodic solutions to (1.1). More precisely, existence and uniqueness of time periodic solutions v per are established by decay properties of solutions operator and the contracting mapping principle, provided that the norm of h j is suitably small. For the details, we refer to Theorem 1. Moreover, the stability of the time periodic solution v per can be studied by investigating the following initial value problem for (1.1) with the initial value when the initial data is small perturbation of the time periodic solution for some fixed initial time t 0 ∈ R. For the details, we refer to Theorem 2.
The study of the global existence and asymptotic behavior of solutions to nonlinear evolution equations has a long history and lots of interesting results have been established. We may refer to [2,3,9,10,11,12,14] and the references therein.
The paper is organized as follows. In Section 2, the decay properties of solution operator to (1.1) are obtained. In Section 3, we prove existence and uniqueness of time periodic solutions to (1.1). Finally, we establish stability of time periodic solutions under suitable conditions in Section 4.
Notations. We give some notations which are used in this paper. Let F[u] denote the Fourier transform of u defined bŷ and we denote its inverse transform by F −1 .

Decay property of solution operator
The aim of this section is to establish decay properties of solution operator to the problem (1.1). We first investigate the linear equation of (1.1): Taking the Fourier transform, we have The characteristic equation of (2.1) is Let λ(ξ) be the corresponding eigenvalues of (2.2), we obtain where F −1 denotes the inverse Fourier transform. The decay estimates of the solution operators S(t) appearing in the solution formula (2.4) is stated as follows.
Lemma 1. Let 1 ≤ p ≤ 2, and let k, κ and l be nonnegative integers. Then we have for 0 ≤ κ ≤ k and φ ∈ W κ,p H k+l .

Existence and uniqueness of time periodic solutions
The purpose of this section is to establish existence and uniqueness of time periodic solutions to the problem (1.1), which has the same period as h j (j = 1, . . . , n). To prove existence and uniqueness of time periodic solutions, we need the following Lemma that has been established in [5] and [19].
for a positive constant M 0 . Let 1 ≤ p, q, r ≤ +∞ and 1 p = 1 q + 1 r , and let k ≥ 0 be an integer. Then we have Our existence and uniqueness of time periodic solutions results are stated as follows: Theorem 1. Let n ≥ 3 and m > n/2 be integers. For any j = 1, . . . , n, assume Proof. The proof of Theorem 1 is divided into two steps. The first step is to prove that the solution to the problem (1.1) is periodic solution, provided that there exists a unique solution to the problem (1.1) . The second step is to prove the problem (1.1) admits a unique solution.
Step 1: If there exists a unique solution to the problem (1.1), this unique solution must be time periodic solution. To this end, we define the following integral equation Noting that n ≥ 3, (2.5) entails that (2.8) and (2.9) imply that which together with n 4 + 1 2 > 1(n ≥ 3) entails the converge of the integral in Then v per 2 (t) is also fixed point. Due to uniqueness of the fixed point, we have v per Y.X. Wang which implies v per 1 (t) is a periodic function with period T . Therefore, the fixed point of the mapping M is a periodic function with period T , provided that the mapping M has a unique fixed point.
Step 2: The problem (1.1) admits a unique solution.
In this step, we shall prove that existence of solutions to the problem (1.1) in the function space C([0, T ]; H m (R n )) by the contraction mapping theorem. To this end, define the function space To prove that there exists a unique solution to the problem (1.1), it is suffice to prove that M has a unique fixed point in the function space Y . For ∀v per ∈ Y and 0 ≤ k ≤ m, it follows from (3.5) and Minkowski inequality that, (2.8), (2.9), Lemma 2 and Sobolev embedding theorem entails that Combining above three estimates gives Taking R = 4CE 0 and letting E 0 be suitably small, we have Finally, we prove M is a strictly contracting mapping. ∀ṽ per ,v per ∈ Y , owing to (3.5), it holds that By (3.7), Minkowski inequality, (2.8), (2.9), Lemma 2 and Sobolev embedding theorem, we obtain Recalling that R = 4CE 0 and letting E 0 suitably small and combining (3.8) yields (3.6) and (3.9) imply that M is a strictly contracting mapping. Consequently, we conclude that there exists a unique fixed point v per ∈ Y of the mapping M, which is a unique solution to (1.1).
Step 1 and Step 2 entails that the problem (1.1) exists a unique time periodic solution. We have complete the proof of Theorem 1.

Remark 1.
Due to the integral with respect to time, in this paper, we only prove that existence and uniqueness of time periodic solutions when the space dimension n ≥ 3. We shall discuss existence and uniqueness of time periodic solutions for n = 1, 2.

Stability of time periodic solutions
In this section, we shall prove the stability of time periodic solutions established in Theorem 1. The asymptotic stability of time periodic solutions is stated as follows.
Theorem 2. Assume the conditions of Theorem 1 hold and v 0 ∈ H m L 1 . Put Then there exists a positive constant δ 1 such that if E 1 ≤ δ 1 , the problem (1.1), (1.2) has a unique global solution v ∈ C([t 0 , ∞); H m (R n )). Moreover, Remark 2. Notice that for time periodic force ∂ xj h j (x, t)(j = 1, . . . , n), there is no time decay in the forcing term so that the large time behavior is more complicated. The method used in this paper relies on the convergence of the integral defined in (3.4). So this convergence can be proved when the space dimension n ≥ 3. Therefore, it is still an open problem for n ≤ 2.
Proof. Firstly, we shall prove the global existence of the solution to the problem (1.1)-(1.2). Without loss of generality, we can assume t 0 = 0. Let v per be the periodic solution constructed in Theorem 1 and let v be a solution to the initial value problem (1.1)-(1.2). Then V = v − v per satisfies the following initial value problem The existence and uniqueness of local solutions may be established by the contraction mapping principle (cf. [3,12,14]). In what follow, global existence of solutions to the problem (1.1)-(1.2) will be proved by continuous argument.
To this end, we assume that where T 0 is the maximal time of existence of local solutions. We may transform the problem (4.2)-(4.3) into the following integral equation

Equation (4.4) and the Minkowski inequality entails that
Making use of (2.5), we have Y.X. Wang It follows from (2.8), Lemma 2 and Sobolev embedding theorem, Theorem 1 that Using (2.9) and the same procedure leading to (4.7), it holds that We insert (4.6)-(4.8) into (4.5) and obtain Letting C 0 = 4C 1 , (4.9) implies that M ≤ C 0 E 1 , provided that δ 0 and E 1 are suitably small. By standard continuous argument, we conclude that the problem (4.2)-(4.3) admits a unique global solution V . Therefore, the problem (1.1)-(1.2) admits a unique global solution v.

Y.X. Wang
Therefore, we arrive at E(t) ≤ CE 1 , provided that δ 0 and E 1 are suitably small. Theorem 2 is proved.