Time-Periodic Solution of the Weakly Dissipative Camassa-Holm Equation

Abstract

This paper is concerned with time-periodic solution of the weakly dissipative Camassa-Holm equation with a periodic boundary condition. The existence and uniqueness of a time periodic solution is presented.

1. Introduction

The Camassa-Holm equation𝑢_𝑡−𝑢_𝑡𝑥𝑥+3𝑢𝑢_𝑥=2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥,𝑡>0,𝑥∈ℝ,(1.1) modeling the unidirectional propagation of shallow water waves over a flat bottom, where 𝑢(𝑡,𝑥) represents the fluid’s free surface above a flat bottom (or equivalently, the fluid velocity at time 𝑡≥0 and in the spatial 𝑥 direction).

Since the equation was derived physically by Camassa and Holm [1, 2], many researchers have paid extensive attention to it. The Camassa-Holm equation is also a model for the propagation of axially symmetric waves in hyperelastic rods [3, 4]. It has a bi-Hamiltonian structure [5, 6] and is completely integrable [1, 2, 7–11]. It is a reexpression of geodesic flow on the diffeomorphism group of the circle [12] and on the Virasoro group [13]. Its solitary waves are peaked [7], and they are orbitally stable and interact like solitons [14–16]. The peakons capture a characteristic of the traveling waves of greatest height-exact traveling solutions of the governing equations for water waves with a peak at their crest [17–19].

The Cauchy problem of the Camassa-Holm equation has been extensively studied. It has been shown that this equation is locally well posed [20–25] for initial data 𝑢₀∈𝐻^𝑠(ℝ) with 𝑠>3/2. Moreover, it has global strong solutions modeling permanent waves [20, 24–27] but also blow-up solutions modeling wave breaking [20–28]. On the other hand, it has global weak solutions with initial data 𝑢₀∈𝐻¹ [29–35]. Moreover, the initial-boundary value problem for the Camassa-Holm equation on the half-line and on a finite interval was discussed in [36, 37]. It is observed that if 𝑢 is the solution of the Camassa-Holm equation with the initial data 𝑢₀ in 𝐻¹(ℝ), we have for all 𝑡>0,‖𝑢(𝑡,⋅)‖_𝐿^∞≤√22‖𝑢(𝑡,⋅)‖^𝐻1≤√22‖‖𝑢⁰(⋅)‖‖_𝐻1.(1.2)

It is worth pointing out that the advantage of the Camassa-Holm equation in comparison with the KdV equation lies in the fact that the Camassa-Holm equation has peaked solitons and models wave breaking [2, 20, 21].

In general, it is difficult to avoid energy dissipation mechanisms in a real world. Ott and Sudan [38] investigated how the KdV equation was modified by the presence of dissipation and the effect of such dissipation on the solitary solution of the KdV equation. Ghidaglia [39] investigated the long-time behavior of solutions to the weakly dissipative KdV equation as a finite-dimensional dynamical system.

The Camassa-Holm equation with dissipative term is𝑢_𝑡−𝑢_𝑡𝑥𝑥+3𝑢𝑢_𝑥−2𝑢_𝑥𝑢_𝑥𝑥−𝑢𝑢_𝑥𝑥𝑥+𝐿(𝑢)=𝑓(𝑡,𝑥),𝑡>0,𝑥∈ℝ,(1.3) where 𝑓(𝑡,𝑥) is the forcing term, 𝐿(𝑢) is a dissipative term, 𝐿 can be a differential operator or a quasi-differential operator according to different physical situations.

With 𝑓=0 and 𝐿(𝑢)=𝛾(1−𝜕^2𝑥)𝑢, (1.3) becomes weakly dissipative Camassa-Holm equation𝑢_𝑡−𝑢_𝑡𝑥𝑥+3𝑢𝑢_𝑥+𝛾𝑢−𝑢_𝑥𝑥=2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥,𝑡>0,𝑥∈ℝ,(1.4) where 𝛾>0 is a constant.

The local well-posedness, global existence, and blow-up phenomena of the Cauchy problem of (1.4) on the line [40] and on the circle [41] were studied. A new global existence result and a new blow-up result for strong solutions to this equation with certain profiles are presented recently [42]. We found that the behaviors of (1.4) are similar to the Camassa-Holm equation in a finite interval of time, such as the local well-posedness and the blow-up phenomena, and that there are considerable differences between (1.4) and the Camassa-Holm equation in their long-time behaviors. The global solutions of (1.4) decay to zero as time goes to infinity provided the potential 𝑦₀=(1−𝜕^2𝑥)𝑢₀ is of one sign (see [40, 41]). This long-time behavior is an important feature that the Camassa-Holm equation does not possess. It is well known that the Camassa-Holm equation has peaked traveling wave solutions. But the fact that any global solution of (1.4) decays to zero means that there are no traveling wave solutions of (1.4).

Another difference between (1.4) and the Camassa-Holm equation is that (1.4) does not have the following conservation laws𝐼₁=_𝑆𝑢𝑑𝑥,𝐼₂=_𝑆𝑢²+𝑢^2𝑥𝑑𝑥,(1.5) which play an important role in the study of the Camassa-Holm equation.

Equation (1.4) has the same blow-up rate as the Camassa-Holm equation does when the blow-up occurs [41]. This fact shows that the blow-up rate of the Camassa-Holm equation is not affected by the weakly dissipative term, but the occurrence of blow-up of (1.4) is affected by the dissipative parameter [40, 41].

In the paper, we would like to consider the following weakly dissipative Camassa-Holm equation𝑢_𝑡−𝑢_𝑡𝑥𝑥+3𝑢𝑢_𝑥−𝑢𝑢2(𝑢(𝑡𝑡_𝑥,𝑢+𝑥_𝑥𝑥+𝜔−,𝐿𝑥𝑢))𝑢==𝑢_𝑥𝑥𝑥𝑢(+(𝑡𝑡,,𝛾𝑥𝑥))𝑢,,−𝑡𝑢𝑡>>_𝑥𝑥00=,,𝑥𝑥𝑓∈∈(𝑡,ℝℝ𝑥,,),𝑡>0,𝑥∈ℝ,(((111...678))) where 𝛾(1−𝜕^2𝑥)𝑢 is the weakly dissipative term, 𝛾>0 is a constant, and the forcing term 𝑓 is 𝜔-periodic in time 𝑡 and 𝐿-periodic in spatial 𝑥. Without loss of generality, we assume further ∫_Ω𝑓(𝑡,𝑥)𝑑𝑥=0, where Ω=[0,𝐿]. When system is periodically dependent on time 𝑡, we want to know whether there exists time-periodic solution with the same period for the system. In many nonlinear evolution equations, the study of time-periodic solution has attracted considerable interest (e.g., [43–45]). In this paper, we will prove that (1.6)–(1.8) have a solution by using the Galerkin method [46], and Leray-Schauder fixed point theorem [44].

Our paper is organized as follows. In Section 2, we give some notations and definition of some space used in this paper. In Section 3, we prove the existence of the approximate solution and give uniform a priori estimates needed where we prove the convergence of a sequence of the approximate solution. Section 4 is devoted to the study of the existence and uniqueness of time-periodic solution for (1.6)–(1.8).

2. Preliminaries

Before starting our work, it is appropriate to introduce some notations and inequalities that will be used in the paper.

Let 𝑋 be a Banach space, we denote by 𝐶^𝑘(𝜔;𝑋) the set of 𝜔-periodic 𝑋-valued measurable functions on ℝ¹ with continuous derivatives up to order 𝑘. The norm in the space 𝐶^𝑘(𝜔;𝑋) is ‖𝑢‖_𝐶^𝑘_(𝜔;𝑋)=sup_{0≤𝑡≤𝜔}{∑^𝑘𝑖=0‖𝐷^𝑖𝑡𝑢‖_𝑋}.

For 1≤𝑝≤∞, the space 𝐿^𝑝(𝜔;𝑋) is the set of 𝜔-periodic 𝑋-valued measurable functions on ℝ such that ‖𝑢‖_𝐿^𝑝_(𝜔;𝑋)=⎧⎪⎨⎪⎩^𝜔0sup_{0≤𝑡≤𝜔}‖𝑢‖^𝑝𝑋𝑑𝑡^1/𝑝<∞,1≤𝑝<∞‖𝑢‖_𝑋<∞,𝑝=∞.(2.1)

The space 𝑊^𝑘,𝑝(𝜔;𝑋) denote the set of functions which belong to 𝐿^𝑝(𝜔;𝑋) together with their derivatives up to order 𝑘, and we write 𝑊^𝑘,2(𝜔;𝑋)=𝐻^𝑘(𝜔;𝑋) in particular when 𝑋 is a Hilbert space.

𝐿^𝑝(Ω) and 𝐻^𝑚(Ω) are classical Sobolev spaces. For simplicity, we write ‖⋅‖_𝐿^𝑝_(Ω) by ‖⋅‖_𝑝 as 𝑝≠2 and ‖⋅‖_𝐿²_(Ω) by ‖⋅‖.

The following inequalities (see [47]) will be used in the proofs later‖‖𝐷‖𝑢‖_∞≤𝑘₁‖𝑢‖_𝐻¹.(2.2)_𝑗𝑢‖‖_𝑝≤𝑘₂‖𝑢‖^𝜃𝐻_𝑚‖𝑢‖^1−𝜃,(2.3) where 𝐷^𝑗𝑢=(𝜕^𝑗𝑢)/(𝜕𝑥^𝑗), 1/𝑝=𝑗+𝜃(1/2−𝑚)+(1−𝜃)(1/2) as 0≤𝑗<𝑚, 𝑗/𝑚≤𝜃≤1.‖𝑢‖≤𝑘₃‖‖𝑢_𝑥‖‖,_Ω𝑢(𝑥)𝑑𝑥=0.(2.4)

3. A Priori Estimates

In this section, we first prove that (1.6)–(1.8) have a sequence of approximate solutions {𝑢_𝑛}^∞𝑛=1, then give a prior, estimates about {𝑢_𝑛}^∞𝑛=1.

We denote the unbounded linear operator 𝐴𝑢=−𝑢_𝑥𝑥 on 𝑋=𝐿²∩{𝑢∣𝑢(𝑥+𝐿)=𝑢(𝑥),∫_Ω𝑢𝑑𝑥=0}, then the set of all linearly independent eigenvectors {𝑤_𝑗}^∞𝑗=0 of 𝐴, that is, 𝐴𝑤_𝑗=𝜆_𝑗𝑤_𝑗, with 0<𝜆₁≤𝜆₂≤⋯≤𝜆_𝑗→∞, is an orthonormal basis of 𝐿²(Ω). For any 𝑛 and a group of function {𝑎_𝑗𝑛(𝑡)}^𝑛𝑗=1, where 𝑎_𝑗𝑛(𝑡)(𝑗=1,2,…,𝑛)∈𝐶¹(𝜔;ℝ), the function 𝑢_𝑛=∑^𝑛𝑗=1𝑎_𝑗𝑛(𝑡)𝑤_𝑗∈𝐶¹(𝜔;𝐻_𝑛) is called an approximate solution to (1.6)–(1.8) if it satisfies the equation as follows:𝑢_𝑛𝑡−𝑢_{𝑛𝑥𝑥𝑡}+𝛾𝑢_𝑛−𝑢_𝑛𝑥𝑥,𝑤_𝑗=𝑁𝑢_𝑛+𝑓,𝑤_𝑗,𝑗=1,…,𝑛,(3.1) where 𝑁𝑢_𝑛=−3𝑢_𝑛𝑢_𝑛𝑥+2𝑢_𝑛𝑥𝑢_𝑛𝑥𝑥+𝑢_𝑛𝑢_{𝑛𝑥𝑥𝑥} and 𝐻_𝑛=span{𝑤₁,𝑤₂,…,𝑤_𝑛}. By the classical theory of ordinary differential equations, for any fixed 𝑣_𝑛(𝑡)=∑^𝑛𝑗=1𝑏_𝑗𝑛(𝑡)𝑤_𝑗∈𝐶¹(𝜔;𝐻_𝑛), the equation (𝑢_𝑛𝑡−𝑢_{𝑛𝑥𝑥𝑡}+𝛾(𝑢_𝑛−𝑢_𝑛𝑥𝑥),𝑤_𝑗)=(𝑁𝑣_𝑛+𝑓,𝑤_𝑗), 𝑗=1,…,𝑛 has a unique 𝜔-periodic solution 𝑢_𝑛 and the mapping 𝐹∶𝑣_𝑛→𝑢_𝑛 is continuous and compact in 𝐶¹(𝜔;𝐻_𝑛). Hence by Leray-Schauder fixed point theorem, we want to prove the existence of an approximate solution only to show sup_{0≤𝑡≤𝜔}‖𝑢_𝑛‖²≤𝑐 for all possible solution of (3.1) replaced by 𝜆𝑁𝑢_𝑛(0≤𝜆≤1) instead of nonlinear term 𝑁𝑢_𝑛, where 𝑐 is a constant which only depends on 𝐿, 𝜀, 𝜔, 𝛾, and 𝑓.

Lemma 3.1. If 𝑓∈𝐶¹(𝜔;𝐻⁻¹(Ω)), then sup_{0≤𝑡≤𝜔}‖‖𝑢^𝑛‖‖²+‖‖𝑢_𝑛𝑥‖‖²≤𝑐¹,(3.2) where 𝑐₁ is a constant which only depends on 𝐿, 𝜔, 𝜀, 𝛾, 𝑘₃, and 𝑓, 𝑀=sup_{0≤𝑡≤𝜔}{‖𝑓(𝑡,𝑥)‖^2𝐻_−1(Ω)} and 𝑑₁=min{2𝛾,2𝛾−𝜀}>0.

Proof. Multiplying (3.1) by 𝑎_𝑗𝑛(𝑡) and summing up over 𝑗 from 1 to 𝑛, we obtain 𝑢_𝑛𝑡−𝑢_{𝑛𝑥𝑥𝑡}+𝛾𝑢_𝑛−𝑢_𝑛𝑥𝑥,𝑢_𝑛=𝑁𝑢_𝑛+𝑓,𝑢_𝑛.(3.3)
Then, we can get 12𝑑𝑑𝑡‖‖𝑢^𝑛‖‖²+‖‖𝑢_𝑛𝑥‖‖²+𝛾‖‖𝑢^𝑛‖‖²+‖‖𝑢_𝑛𝑥‖‖²=𝑁𝑢^𝑛+𝑓,𝑢_𝑛.(3.4)
Notice that −3∫_Ω𝑢^2𝑛𝑢_𝑛𝑥𝑑𝑥=0, 2∫_Ω𝑢_𝑛𝑢_𝑛𝑥𝑢_𝑛𝑥𝑥𝑑𝑥+∫_Ω𝑢^2𝑛𝑢_{𝑛𝑥𝑥𝑥}𝑑𝑥=0.
From Young’s inequality, we have ∫_Ω𝑓𝑢_𝑛𝑑𝑥≤(𝜀/2)‖𝑢_𝑛𝑥‖²+𝑘²³𝑀/2𝜀, where 𝜀>0 is a constant.
According to the above relations, we can derive from (3.4) that 𝑑𝑑𝑡‖‖𝑢^𝑛‖‖²+‖‖𝑢_𝑛𝑥‖‖²+𝑑¹‖‖𝑢^𝑛‖‖²+‖‖𝑢_𝑛𝑥‖‖²≤𝑘²³𝑀𝜀,(3.5) where 𝑑₁=min{2𝛾,2𝛾−𝜀}>0.
Considering the time periodicity of 𝑢_𝑛 and integrating (3.5) over [0,𝜔], we get 𝑑₁^𝜔0‖‖𝑢^𝑛‖‖²+‖‖𝑢_𝑛𝑥‖‖²𝑑𝑡≤𝜔𝑘²³𝑀𝜀.(3.6)
Hence, there exists 𝑡^∗∈[0,𝜔) such that ‖𝑢_𝑛(𝑡^∗)‖²+‖𝑢_𝑛𝑥(𝑡^∗)‖²≤𝑘²³𝑀/𝑑₁𝜀.
From (3.5), we have (𝑑/𝑑𝑡)(‖𝑢_𝑛‖²+‖𝑢_𝑛𝑥‖²)≤𝑘²³𝑀/𝜀.
Integrating the above inequality with respect to 𝑡 from 𝑡^∗ to 𝑡∈[𝑡^∗,𝑡^∗+𝜔], we deduce that ‖‖𝑢_𝑛(𝑡)‖‖²+‖‖𝑢_𝑛𝑥(𝑡)‖‖²≤‖‖𝑢_𝑛(𝑡^∗)‖‖²+‖‖𝑢_𝑛𝑥(𝑡^∗)‖‖²+𝜔𝑘²³𝑀𝜀≤𝑑1₁+𝜔𝑘²³𝑀𝜀.(3.7)
Hence, we infer sup_{0≤𝑡≤𝜔}‖‖𝑢^𝑛‖‖²+‖‖𝑢_𝑛𝑥‖‖²≤1𝑑₁+𝜔𝑘²³𝑀𝜀≜𝑐¹,(3.8) which concludes our proof.

From Lemma 3.1 and Leray-Schauder fixed point theorem, (3.1) has solution {𝑢_𝑛}^∞𝑛=1, which is also a sequence of approximate solutions of (1.6)–(1.8). In order to obtain the convergence of sequence {𝑢_𝑛}^∞𝑛=1, we need to give a priori estimates for the high-order derivers of {𝑢_𝑛}^∞𝑛=1.

Lemma 3.2. If 𝑓∈𝐶¹(𝜔;𝐻⁻¹(Ω)), then sup_{0≤𝑡≤𝜔}‖‖𝑢^𝑛𝑥‖‖²+‖‖𝑢_𝑛𝑥𝑥‖‖²≤𝑐²,(3.9) where 𝑐₂ is a constant which only depends on 𝐿, 𝜔, 𝜀, 𝛾, 𝜆_𝑛, 𝑘₁, 𝑘₂, 𝑘₃, and 𝑓, 𝑀=sup_{0≤𝑡≤𝜔}{‖𝑓(𝑡,𝑥)‖^2𝐻_−1(Ω)} and 𝑑₂=min{2𝛾−(13/2)𝜀𝜆_𝑛,2𝛾−(21/2)𝜀}>0.

Proof. Multiplying (3.1) by −𝜆_𝑗𝑎_𝑗𝑛(𝑡) and summing up over 𝑗 from 1 to 𝑛, we have 𝑢_𝑛𝑡−𝑢_{𝑛𝑥𝑥𝑡}+𝛾𝑢_𝑛−𝑢_𝑛𝑥𝑥,𝑢_𝑛𝑥𝑥=𝑁𝑢_𝑛+𝑓,𝑢_𝑛𝑥𝑥.(3.10)
The above equation yields −12𝑑𝑑𝑡‖‖𝑢^𝑛𝑥‖‖²+‖‖𝑢_𝑛𝑥𝑥‖‖²−𝛾‖‖𝑢^𝑛𝑥‖‖²+‖‖𝑢_𝑛𝑥𝑥‖‖²=𝑁𝑢^𝑛+𝑓,𝑢_𝑛𝑥𝑥.(3.11)
From Young’s inequality, we have ||||_Ω𝑓𝑢_𝑛𝑥𝑥𝑑𝑥||||≤𝜀‖‖𝑢_{𝑛𝑥𝑥𝑥}‖‖²+𝑘²³𝑀4𝜀,(3.12) where 𝜀>0 is a constant.
From (2.2), (3.8), and Young’s inequality, we can deduce that ||||_Ω𝑢_𝑛𝑢_𝑛𝑥𝑢_𝑛𝑥𝑥𝑑𝑥||||≤‖‖𝑢_𝑛‖‖_∞_Ω≤𝑘₁||𝑢_𝑛𝑥𝑢_𝑛𝑥𝑥||𝑑𝑥‖‖𝑢_𝑛‖‖_𝐻1_Ω≤𝑘₁||𝑢_𝑛𝑥𝑢_𝑛𝑥𝑥||𝑑𝑥𝑐^11/2𝑘₁𝜀𝑐^11/2‖‖𝑢_𝑛𝑥𝑥‖‖²+𝑘¹𝑐^11/24𝜀‖‖𝑢^𝑛𝑥‖‖²≤𝜀‖‖𝑢_𝑛𝑥𝑥‖‖²+𝑘²¹𝑐²¹4𝜀.(3.13)
From (2.3), (3.8), Cauchy-Schwarz inequality, Young’s inequality, and Lemma 3.1, we get ||||_Ω𝑢_𝑛𝑢_𝑛𝑥𝑥𝑢_{𝑛𝑥𝑥𝑥}𝑑𝑥||||=||||−12_Ω𝑢_𝑛𝑥𝑢^{2𝑛𝑥𝑥}𝑑𝑥||||≤12‖‖𝑢^𝑛𝑥‖‖‖‖𝑢_𝑛𝑥𝑥‖‖²⁴≤12𝑐^11/2𝑘²²‖‖𝑢_𝑛‖‖^1/2‖‖𝑢_𝑛‖‖^𝐻3/2₃≤34𝜀‖‖𝑢^𝑛‖‖^2𝐻₃+𝑐³¹𝑘⁸²64𝜀³≤34𝜀‖‖𝑢^𝑛𝑥𝑥‖‖²+34𝜀‖‖𝑢^{𝑛𝑥𝑥𝑥}‖‖²+34𝜀𝑐¹+𝑐³¹𝑘⁸²64𝜀³‖‖𝑢,(3.14)_{𝑛𝑥𝑥𝑥}‖‖²=_Ω|||||(_𝑛_𝑗=1𝑎_𝑗𝑛(𝑡)𝑤_𝑗)_𝑥𝑥𝑥|||||²𝑑𝑥=_Ω|||||(_𝑛_𝑗=1𝜆_𝑗𝑎_𝑗𝑛(𝑡)𝑤_𝑗)_𝑥|||||²𝑑𝑥≤𝜆_𝑛‖‖𝑢_𝑛𝑥‖‖².(3.15)
Taking (3.11)–(3.15) into account, we can infer that 𝑑𝑑𝑡‖‖𝑢^𝑛𝑥‖‖²+‖‖𝑢_𝑛𝑥𝑥‖‖²+𝑑²‖‖𝑢^𝑛𝑥‖‖²+‖‖𝑢_𝑛𝑥𝑥‖‖²≤𝑘²³𝑀2𝜀+3𝑘²¹𝑐²¹2𝜀+9𝜀2𝑐¹+3𝑐³¹𝑘⁸²32𝜀³,(3.16) where 𝑑₂=min{2𝛾−(13/2)𝜀𝜆_𝑛,2𝛾−(21/2)𝜀}>0.
Integrating (3.16) about 𝑡 from 0 to 𝜔 and considering the time periodicity of 𝑢_𝑛, we get 𝑑₂^𝜔0‖‖𝑢^𝑛𝑥‖‖²+‖‖𝑢_𝑛𝑥𝑥‖‖²𝑑𝑡≤𝑘²³𝑀2𝜀+3𝑘²¹𝑐²¹2𝜀+9𝜀2𝑐¹+3𝑐³¹𝑘⁸²32𝜀³𝜔.(3.17)
Hence, there exists 𝑡^∗∈[0,𝜔) such that ‖‖𝑢_𝑛𝑥(𝑡^∗)‖‖²+‖‖𝑢_𝑛𝑥𝑥(𝑡^∗)‖‖²≤1𝑑₂𝑘²³𝑀2𝜀+3𝑘²¹𝑐²¹2𝜀+9𝜀2𝑐¹+3𝑐³¹𝑘⁸²32𝜀³.(3.18)
From (3.16), we have ‖‖𝑢_𝑛𝑥(𝑡)‖‖²+‖‖𝑢_𝑛𝑥𝑥(𝑡)‖‖²≤‖‖𝑢_𝑛𝑥(𝑡^∗)‖‖²+‖‖𝑢_𝑛𝑥𝑥(𝑡^∗)‖‖²+𝑘²³𝑀2𝜀+3𝑘²¹𝑐²¹2𝜀+9𝜀2𝑐¹+3𝑐³¹𝑘⁸²32𝜀³𝜔.(3.19)
Then we can obtain sup_{0≤𝑡≤𝜔}‖‖𝑢^𝑛𝑥‖‖²+‖‖𝑢_𝑛𝑥𝑥‖‖²≤1𝑑₂+𝜔𝑘²³𝑀2𝜀+3𝑘²¹𝑐²¹2𝜀+9𝜀2𝑐¹+3𝑐³¹𝑘⁸²32𝜀³≜𝑐²,(3.20) which concludes our proof.

In the following, we continue to establish a priori estimates for high-order derivers of the approximate solution {𝑢_𝑛}^∞𝑛=1 by an inductive argument.

Lemma 3.3. For any 𝑘≥0, if 𝑓∈𝐶¹(𝜔;𝐻^𝑘−1(Ω)), then sup_{0≤𝑡≤𝜔}‖‖𝐷^𝑘+1𝑢_𝑛‖‖²+‖‖𝐷^𝑘+2𝑢_𝑛‖‖²≤𝑐,(3.21) where 𝑐 is a constant which only depends on 𝐿, 𝜔, 𝜀, 𝛾, 𝜆_𝑛, 𝑘, 𝑘₁, 𝑘₂, 𝑘₃, 𝑓 and 𝑑₃={2𝛾−16𝜀𝜆_𝑛,2𝛾−14𝜀}>0.

Proof. By Lemma 3.2, we know the conclusion of Lemma 3.3 holds for 𝑘=0.
Assume that for 𝑘≤𝑚−1(𝑚≥2) the conclusion of Lemma 3.3 holds, we want to prove that the same statement holds for 𝑘=𝑚 also.
Multiplying (3.1) by (−1)^𝑚+1𝜆^𝑗𝑚+1𝑎_𝑗𝑛(𝑡) and summing up over 𝑗 from 1 to 𝑛, we have (−1)^𝑚+112𝑑𝑑𝑡‖‖𝐷^𝑚+1𝑢_𝑛‖‖²+‖‖𝐷=^𝑚+2𝑁𝑢_𝑛𝑢‖‖²_𝑛+(−1)^𝑚+1𝛾‖‖𝐷^𝑚+1𝑢_𝑛‖‖²+‖‖𝐷^𝑚+2𝑢_𝑛‖‖²+𝑓,𝐷^2(𝑚+1)𝑢_𝑛.(3.22)
Follow the same methods discussed in Lemma 3.2, we have ||||_Ω𝑓𝐷^2(𝑚+1)𝑢_𝑛𝑑𝑥||||=||||_Ω𝐷^𝑚−1𝑓𝐷^𝑚+3𝑢_𝑛𝑑𝑥||||≤𝜀‖‖𝐷^𝑚+3𝑢_𝑛‖‖²+14𝜀‖‖𝐷^𝑚−1𝑓‖‖².(3.23)
From the conclusion of Lemma 3.3 for 𝑘≤𝑚−1, (2.2), (2.4) and Young’s inequality, we can deduce that ||||_Ω𝑢_𝑛𝑢_𝑛𝑥𝐷^2(𝑚+1)𝑢_𝑛𝑑𝑥||||=|||||_Ω^𝑚+1_𝑖=0𝐶^𝑖𝑚+1𝐷^𝑖𝑢_𝑛𝐷^{𝑚+1−𝑖}𝑢_𝑛𝑥𝐷^𝑚+1≤𝑢_𝑛𝑑𝑥_Ω|||||+||𝑢_𝑛𝐷^𝑚+2𝑢_𝑛𝐷^𝑚+1𝑢_𝑛||𝑑𝑥_Ω|||||_𝑚+1_𝑖=1𝐶^𝑖𝑚+1𝐷^𝑖𝑢_𝑛𝐷^{𝑚+1−𝑖}𝑢_𝑛𝑥𝐷^𝑚+1𝑢_𝑛≤𝜀‖‖𝐷_𝑚+2|||||𝑑𝑥≤𝜀‖‖𝐷_𝑚+2𝑢_𝑛‖‖²+𝑐𝜀,𝑘₁,𝑘₃‖‖𝐷^𝑚+1𝑢_𝑛‖‖²+𝑐𝑚,𝑘₁𝑢_𝑛‖‖²+𝑐𝜀,𝑘₁,𝑘₃,𝑚.(3.24)
Similarly, we can also deduce that ||||_Ω𝑢_𝑛𝑢_{𝑛𝑥𝑥𝑥}𝐷^2(𝑚+1)𝑢_𝑛𝑑𝑥||||=|||||_Ω^𝑚_𝑖=0𝐶^𝑖𝑚𝐷^𝑖𝑢_𝑛𝐷^𝑚−𝑖𝑢_{𝑛𝑥𝑥𝑥}𝐷^𝑚+2≤𝑢_𝑛𝑑𝑥_Ω|||||+||𝑢_𝑛𝐷^𝑚+3𝑢_𝑛𝐷^𝑚+2𝑢_𝑛||𝑑𝑥+_Ω|||𝐶_1𝑚𝐷𝑢_𝑛𝐷^𝑚+2𝑢_𝑛²|||𝑑𝑥_Ω+||𝐶^2𝑚𝐷²𝑢_𝑛𝐷^𝑚+1𝑢_𝑛𝐷^𝑚+2𝑢_𝑛||𝑑𝑥_Ω|||||_𝑚_𝑖=3𝐶^𝑖𝑚𝐷^𝑖𝑢_𝑛𝐷^{𝑚+3−𝑖}𝑢_𝑛𝐷^𝑚+2𝑢_𝑛≤‖‖𝑢|||||𝑑𝑥_𝑛‖‖_∞_Ω||𝐷^𝑚+3𝑢_𝑛𝐷^𝑚+2𝑢_𝑛||𝑑𝑥+𝑚‖‖𝐷𝑢_𝑛‖‖_∞_Ω+𝐶_2𝑚||𝐷^𝑚+2𝑢_𝑛||²𝑑𝑥‖‖𝐷²𝑢_𝑛‖‖_∞_Ω+||𝐷^𝑚+1𝑢_𝑛𝐷^𝑚+2𝑢_𝑛||𝑑𝑥_𝑚_𝑖=3𝐶^𝑖𝑚‖‖𝐷^𝑖𝑢_𝑛‖‖_∞‖‖𝐷^{𝑚+3−𝑖}𝑢_𝑛‖‖_∞_Ω≤𝑐𝑘₁||𝐷^𝑚+2𝑢_𝑛||𝑑𝑥+,𝑘₃,𝑚_Ω||𝐷^𝑚+3𝑢_𝑛𝐷^𝑚+2𝑢_𝑛||𝑑𝑥+_Ω||𝐷^𝑚+2𝑢_𝑛||²𝑑𝑥_Ω||𝐷^𝑚+1𝑢_𝑛𝐷^𝑚+2𝑢_𝑛||𝑑𝑥+_Ω||𝐷^𝑚+2𝑢_𝑛||𝑑𝑥.(3.25)
From the conclusion of Lemma 3.3 for 𝑘≤𝑚−1, Young’s inequality and (2.3), we have 𝑐𝑘₁,𝑘₃,𝑚_Ω||𝐷^𝑚+3𝑢_𝑛𝐷^𝑚+2𝑢_𝑛||𝑑𝑥≤𝜀‖‖𝐷^𝑚+3𝑢_𝑛‖‖²+𝑐𝑘₁,𝑘₃,𝑚,𝜀‖‖𝐷^𝑚+2𝑢_𝑛‖‖²≤×𝜀‖‖‖‖𝐷𝑢^𝑚+3𝑢_𝑛‖‖²+𝑐𝑘₁,𝑘₂,𝑘₃,𝑚,𝜀_𝑛‖‖^{𝐻2(𝑚+2)/𝑚+3}_𝑚+3‖‖𝑢_𝑛‖‖^{2(1−(𝑚+2)/(𝑚+3))}≤𝜀‖‖𝐷^𝑚+3𝑢_𝑛‖‖²+𝜀‖‖𝑢_𝑛‖‖^2𝐻_𝑚+3+𝑐𝑘₁,𝑘₂,𝑘₃,𝑚,𝜀‖‖𝑢_𝑛‖‖²𝑐𝑘≤2𝜀‖‖𝐷^𝑚+3𝑢_𝑛‖‖²+𝜀‖‖𝐷^𝑚+2𝑢_𝑛‖‖²+𝑐𝑘₁,𝑘₂,𝑘₃,𝑚,𝜀₁,,𝑘₃,𝑚_Ω|||𝐷_𝑚+2𝑢_𝑛²|||𝑑𝑥=𝑐𝑘₁,𝑘₃,𝑚‖‖𝐷^𝑚+2𝑢_𝑛‖‖²≤𝑐𝑘₁,𝑘₂,𝑘₃,𝑚‖‖𝑢_𝑛‖‖^{𝐻2(𝑚+2)/(𝑚+3)}_𝑚+3‖‖𝑢_𝑛‖‖^{2(1−(𝑚+2)/(𝑚+3))}𝑐𝑘≤𝜀‖‖𝐷^𝑚+3𝑢_𝑛‖‖²+𝜀‖‖𝐷^𝑚+2𝑢_𝑛‖‖²+𝑐𝑘₁,𝑘₂,𝑘₃,𝑚,𝜀₁,,𝑘₃,𝑚_Ω||𝐷^𝑚+1𝑢_𝑛𝐷^𝑚+2𝑢_𝑛||𝑑𝑥≤𝜀‖‖𝐷^𝑚+2𝑢_𝑛‖‖²+𝑐𝑘₁,𝑘₃,𝑚,𝜀‖‖𝐷^𝑚+1𝑢_𝑛‖‖²𝑐𝑘≤𝜀‖‖𝐷^𝑚+2𝑢_𝑛‖‖²+𝑐𝑘₁,𝑘₃,𝑚,𝜀₁,,𝑘₃,𝑚_Ω||𝐷^𝑚+2𝑢_𝑛||𝑑𝑥≤𝜀‖‖𝐷^𝑚+2𝑢_𝑛‖‖²+𝑐𝑘₁,𝑘₃,𝑚,𝜀,𝐿.(3.26)
Combining (3.25) and the above inequality, we can get ||||_Ω𝑢_𝑛𝑢_{𝑛𝑥𝑥𝑥}𝐷^2(𝑚+1)𝑢_𝑛𝑑𝑥||||≤3𝜀‖‖𝐷^𝑚+3𝑢_𝑛‖‖²+4𝜀‖‖𝐷^𝑚+2𝑢_𝑛‖‖²+𝑐𝑘₁,𝑘₂,𝑘₃,𝑚,𝜀,𝐿.(3.27)
Similarly, ||||_Ω𝑢_𝑛𝑥𝑢_𝑛𝑥𝑥𝐷^2(𝑚+1)𝑢_𝑛𝑑𝑥||||≤+_Ω||𝑢_𝑛𝑥𝐷^𝑚+1𝑢_𝑛𝐷^𝑚+3𝑢_𝑛||𝑑𝑥_Ω|||||_𝑚−1_𝑖=1𝐶^𝑖𝑚−1𝐷^𝑖+1𝑢_𝑛𝐷^{𝑚+1−𝑖}𝑢_𝑛𝐷^𝑚+3𝑢_𝑛≤‖‖𝑢|||||𝑑𝑥_𝑛𝑥‖‖_∞_Ω+||𝐷^𝑚+1𝑢_𝑛𝐷^𝑚+3𝑢_𝑛||𝑑𝑥_𝑚−1_𝑖=1𝐶^𝑖𝑚−1‖‖𝐷^𝑖+1𝑢_𝑛‖‖_∞‖‖𝐷^{𝑚+1−𝑖}𝑢_𝑛‖‖_∞_Ω≤2𝜀‖‖𝐷_𝑚+3||𝐷^𝑚+3𝑢_𝑛||𝑑𝑥𝑢_𝑛‖‖²+𝑐𝑚,𝑘₁,𝜀,𝐿.(3.28)
Taking (3.22)–(3.24) and (3.27)-(3.28) into account, we can deduce that 𝑑𝑑𝑡‖‖𝐷^𝑚+1≤16𝜀‖‖𝑢𝐷^𝑛‖‖²_𝑚+3+‖‖𝐷^𝑚+2𝑢_𝑛‖‖²+2𝛾‖‖𝐷^𝑚+1𝑢_𝑛‖‖²+‖‖𝐷^𝑚+2𝑢_𝑛‖‖²𝑢_𝑛‖‖²+14𝜀‖‖𝐷^𝑚+2𝑢_𝑛‖‖²+𝑐𝑘₁,𝑘₂,𝑘₃,𝑚,𝜀,𝑓,𝐿.(3.29)
From the above relation, we can infer 𝑑𝑑𝑡‖‖𝐷^𝑚+1𝑢_𝑛‖‖²+‖‖𝐷^𝑚+2𝑢_𝑛‖‖²+𝑑³‖‖𝐷^𝑚+1𝑢_𝑛‖‖²+‖‖𝐷^𝑚+2𝑢_𝑛‖‖²≤𝑐𝑘¹,𝑘₂,𝑘₃,𝑚,𝜀,𝑓,𝐿,(3.30) where 𝑑₃={2𝛾−16𝜀𝜆_𝑛,2𝛾−14𝜀}>0.
Integrating (3.30) about 𝑡 from 0 to 𝜔, there exists 𝑡^∗∈[0,𝜔) such that ‖‖𝐷^𝑚+1𝑢_𝑛(𝑡^∗)‖‖²+‖‖𝐷^𝑚+2𝑢_𝑛(𝑡^∗)‖‖²≤𝑐𝑘¹,𝑘₂,𝑘₃,𝑚,𝜀,𝑓,𝐿𝑑₃.(3.31)
From (3.30), we have 𝑑𝑑𝑡‖‖𝐷^𝑚+1𝑢_𝑛‖‖²+‖‖𝐷^𝑚+2𝑢_𝑛‖‖²≤𝑐𝑘¹,𝑘₂,𝑘₃,𝑚,𝜀,𝑓,𝐿.(3.32)
Integrating the above inequality from 𝑡^∗ to 𝑡∈[𝑡^∗,𝑡^∗+𝜔] and with (3.31), we can easily obtain sup_{0≤𝑡≤𝜔}‖‖𝐷^𝑚+1𝑢_𝑛‖‖²+‖‖𝐷^𝑚+2𝑢_𝑛‖‖²≤1𝑑₃+𝜔𝑐𝑘₁,𝑘₂,𝑘₃,𝑚,𝜀,𝑓,𝐿≜𝑐.(3.33)
The proof is completed.

Lemma 3.4. For any 𝑘≥0, if 𝑓∈𝐶¹(𝜔;𝐻^𝑘+1(Ω)), then sup_{0≤𝑡≤𝜔}‖‖𝐷^𝑘𝑢_𝑛𝑡‖‖²+‖‖𝐷^𝑘+1𝑢_𝑛𝑡‖‖²≤𝑐(3.34) where 𝑐 is a constant which only depends on 𝐿, 𝜔, 𝜀, 𝛾, 𝜆_𝑛, 𝑘, 𝑘₁, 𝑘₂, 𝑘₃, and 𝑓.

Proof. We first prove the conclusion of Lemma 3.4 holds for 𝑘=0. Multiplying (3.1) by 𝑎^𝑗𝑛(𝑡) and summing up over 𝑗 from 1 to 𝑛, we have ‖‖𝑢_𝑛𝑡‖‖²+‖‖𝑢_𝑛𝑥𝑡‖‖²=𝑁𝑢_𝑛+𝑓−𝛾𝑢_𝑛−𝑢_𝑛𝑥𝑥,𝑢_𝑛𝑡.(3.35)
By Lemma 3.3, if 𝑓∈𝐶¹(𝜔;𝐻¹(Ω)), then we have ‖𝑢_𝑛‖^2𝐻₄≤𝑐. Hence, ||𝑁𝑢_𝑛+𝑓−𝛾𝑢_𝑛−𝑢_𝑛𝑥𝑥,𝑢_𝑛𝑡||≤‖‖𝑁𝑢_𝑛+𝑓−𝛾𝑢_𝑛−𝑢_𝑛𝑥𝑥‖‖‖‖𝑢_𝑛𝑡‖‖≤𝑐‖‖𝑢_𝑛𝑡‖‖.(3.36)
Therefore, from (3.35) and (3.36), it is easy to know that sup_{0≤𝑡≤𝜔}‖‖𝑢^𝑛𝑡‖‖²+‖‖𝑢_𝑛𝑥𝑡‖‖²≤𝑐.(3.37)
Assume that the conclusion of Lemma 3.4 holds for 𝑘≤𝑚(𝑚≥1), we want to prove that the conclusion of Lemma 3.4 also holds for 𝑘=𝑚+1.
Multiplying (3.1) by (−1)^𝑚+1𝜆^𝑗𝑚+1𝑎^𝑗𝑛(𝑡) and summing up over 𝑗 from 1 to 𝑛, we have (−1)^𝑚+1‖‖𝐷^𝑚+1𝑢_𝑛𝑡‖‖²+‖‖𝐷^𝑚+2𝑢_𝑛𝑡‖‖²=𝑁𝑢^𝑛+𝑓−𝛾𝑢_𝑛−𝑢_𝑛𝑥𝑥,𝐷^2(𝑚+1)𝑢_𝑛𝑡.(3.38)
By Lemma 3.3, if 𝑓∈𝐶¹(𝜔;𝐻^𝑚+2(Ω)), then ‖𝐷^𝑘𝑢_𝑛‖²≤𝑐 for 𝑘≤𝑚+5. Hence, ||𝑁𝑢_𝑛+𝑓−𝛾𝑢_𝑛−𝑢_𝑛𝑥𝑥,𝐷^2(𝑚+1)𝑢_𝑛𝑡||≤≤𝑐‖‖‖‖𝐷𝐷^𝑚+1_𝑚+1𝑁𝑢_𝑛+𝑓−𝛾𝑢_𝑛−𝑢_𝑛𝑥𝑥‖‖‖‖𝐷^𝑚+1𝑢_𝑛𝑡‖‖𝑢_𝑛𝑡‖‖.(3.39)
Taking (3.38) and (3.39) into account, it follows sup_{0≤𝑡≤𝜔}‖‖𝐷^𝑚+1𝑢_𝑛𝑡‖‖²+‖‖𝐷^𝑚+2𝑢_𝑛𝑡‖‖²≤𝑐.(3.40)
This completes the proof of Lemma 3.4 by an inductive argument.

4. Existence and Uniqueness of Time-Periodic Solution

We have proved that (1.6)–(1.8) have a sequence of approximate solutions {𝑢_𝑛}^∞𝑛=1. In this section, we want to prove that the sequence converges and the limit is a solution of (1.6)–(1.8).

By Lemmas 3.1–3.4 and standard compactness arguments, we conclude that there is a subsequence which we denote also by {𝑢_𝑛} such that for any 𝐾≥0, if 𝑓∈𝐶¹(𝜔;𝐻^𝑘+1(Ω)), we have𝑢_𝑛𝑢(𝑡)⟶𝑢(𝑡),weakly∗in𝐿^∞𝜔;𝐻^𝑘+4(Ω)_𝑛,𝑢(𝑡)⟶𝑢(𝑡),stronglyin𝐿^∞𝜔;𝐻^𝑘+3(Ω)_𝑛𝑡,𝑢(𝑡)⟶𝑢_𝑡(𝑡),weakly∗in𝐿^∞𝜔;𝐻^𝑘+1(Ω)_𝑛𝑡,(𝑡)⟶𝑢_𝑡(𝑡),stronglyin𝐿^∞𝜔;𝐻^𝑘(Ω).(4.1)

From the above lemmas, we know that the nonlinear terms are well defined‖‖𝑢_𝑛𝑢_𝑛𝑥−𝑢𝑢_𝑥‖‖≤≤‖‖‖‖𝑢𝑢_𝑛𝑢_𝑛𝑥−𝑢_𝑥‖‖+‖‖𝑢_𝑥𝑢_𝑛−𝑢_𝑛‖‖‖‖_∞‖‖𝑢_𝑛𝑥−𝑢_𝑥‖‖+‖‖𝑢_𝑥‖‖_∞‖‖𝑢_𝑛−𝑢‖‖⟶0,(4.2) as 𝑛→∞, uniformly in 𝑡,‖‖𝑢_𝑛𝑥𝑢_𝑛𝑥𝑥−𝑢_𝑥𝑢_𝑥𝑥‖‖≤≤‖‖‖‖𝑢𝑢_{𝑛𝑥𝑛𝑥}𝑢_𝑛𝑥𝑥−𝑢_𝑥𝑥‖‖+‖‖𝑢_𝑥𝑥𝑢_𝑛𝑥−𝑢_𝑥‖‖‖‖_∞‖‖𝑢_𝑛𝑥𝑥−𝑢_𝑥𝑥‖‖+‖‖𝑢_𝑥𝑥‖‖_∞‖‖𝑢_𝑛𝑥−𝑢_𝑥‖‖⟶0,(4.3) as 𝑛→∞, uniformly in 𝑡,‖‖𝑢_𝑛𝑢_{𝑛𝑥𝑥𝑥}−𝑢𝑢_𝑥𝑥𝑥‖‖≤≤‖‖‖‖𝑢𝑢_𝑛𝑢_{𝑛𝑥𝑥𝑥}−𝑢_𝑥𝑥𝑥‖‖+‖‖𝑢_𝑥𝑥𝑥𝑢_𝑛−𝑢_𝑛‖‖≤‖‖𝑢_𝑛‖‖_∞‖‖𝑢_{𝑛𝑥𝑥𝑥}−𝑢_𝑥𝑥𝑥‖‖+‖‖𝑢_𝑛−𝑢‖‖_∞‖‖𝑢_𝑥𝑥𝑥‖‖‖‖_∞‖‖𝑢_{𝑛𝑥𝑥𝑥}−𝑢_𝑥𝑥𝑥‖‖+𝑘₁‖‖𝑢_𝑛−𝑢‖‖_𝐻1‖‖𝑢_𝑥𝑥𝑥‖‖⟶0,(4.4) as 𝑛→∞, uniformly in 𝑡.

Consequently, it follows that𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝛾𝑢−𝑢_𝑥𝑥,𝜂=(𝑁𝑢+𝑓,𝜂),𝜂∈𝐿²_per.(4.5)

Thanks to the estimates obtained in the previous section, we have𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝛾𝑢−𝑢_𝑥𝑥=𝑁𝑢+𝑓,(4.6) a.e. on ℝ¹×Ω.

So we obtain that the existence of time periodic solution for (1.6)–(1.8), which is the following theorem.

Theorem 4.1. Given 𝑓∈𝐶¹(𝜔;𝐻^𝑘+1(Ω)),  𝑘≥0, there exists a time periodic solution 𝑢(𝑡,𝑥) to (1.6)–(1.8), such that 𝑢(𝑡,𝑥)∈𝐿^∞(𝜔;𝐻^𝑘+4(Ω))∩𝑊^1,∞(𝜔;𝐻^𝑘(Ω)).

Under the assumption of Theorem 4.1, we are unable to prove the uniqueness of the solution for (1.6)–(1.8). But if we assume that 𝑀 is sufficiently small, then the result can be obtained.

Theorem 4.2. Suppose that the assumption in Theorem 4.1 holds. If 𝑀 is sufficiently small, then the time periodic solution of (1.6)–(1.8) in Theorem 4.1 is unique.

Proof. Let 𝑢 and 𝑢 be any two time periodic solutions of (1.6)–(1.8). With 𝑣=𝑢−𝑢, we can get from (1.6) that 𝑣_𝑡−𝑣_𝑥𝑥𝑡+𝛾𝑣−𝑣_𝑥𝑥=𝑁𝑢−𝑁𝑢.(4.7)
Taking the inner product of (4.7) with 𝑣, we have 12𝑑𝑑𝑡‖𝑣‖²+‖‖𝑣_𝑥‖‖²+𝛾‖𝑣‖²+‖‖𝑣_𝑥‖‖²=𝑁𝑢−𝑁𝑢,𝑣.(4.8)
Since, ||−3𝑢𝑢_𝑥+3𝑢𝑢_𝑥,𝑣||≤≤3||−3𝑢𝑣_𝑥,𝑣||+||−3𝑢_𝑥𝑣,𝑣||2‖𝑢‖^∞‖𝑣‖²+‖‖𝑣_𝑥‖‖²+3‖‖𝑢_𝑥‖‖_∞‖𝑣‖²≤3𝑘¹2𝑐^11/2+3𝑘₁𝑐^21/2‖𝑣‖²+3𝑘¹2𝑐^11/2||2𝑢_𝑥‖‖𝑣_𝑥‖‖²,(4.9)𝑢_𝑥𝑥−2𝑢_𝑥𝑢_𝑥𝑥,𝑣||≤≤‖‖||𝑢2𝑢_𝑥𝑣_𝑥𝑥,𝑣||+||2𝑢_𝑥𝑥𝑣_𝑥,𝑣_𝑥𝑥||≤2𝑘₁‖‖_∞‖𝑣‖²+‖‖𝑣_𝑥‖‖²+2‖‖𝑢^𝑥‖‖_∞‖‖𝑣_𝑥‖‖²+‖‖𝑢_𝑥𝑥‖‖_∞‖𝑣‖²+‖‖𝑣_𝑥‖‖²||𝑢𝑢_𝑥𝑥𝑥𝑐₂+𝑐^1/2‖𝑣‖²+2𝑘₁𝑐₂+𝑐^1/2+2𝑘₁𝑐^21/2‖‖𝑣^𝑥‖‖²,(4.10)−𝑢𝑢_𝑥𝑥𝑥,𝑣||≤≤||𝑢𝑣_𝑥𝑥𝑥,𝑣||+||𝑢_𝑥𝑥𝑥𝑣,𝑣_Ω||||𝑢_𝑥𝑥𝑣𝑣_𝑥||𝑑𝑥+32_Ω≤1||𝑢_𝑥𝑣^2𝑥||𝑑𝑥+2_Ω||𝑢_𝑥𝑥𝑣𝑣_𝑥||𝑑𝑥2‖‖𝑢^𝑥𝑥‖‖_∞‖𝑣‖²+‖‖𝑣_𝑥‖‖²+32‖‖𝑢^𝑥≤3𝑘₁‖‖_∞‖‖𝑣_𝑥‖‖²+‖‖𝑢_𝑥𝑥‖‖_∞‖𝑣‖²+‖‖𝑣_𝑥‖‖²2𝑐²+𝑐^1/2‖𝑣‖²+3𝑘¹2𝑐²+𝑐^1/2+3𝑘¹2𝑐^21/2‖‖𝑣^𝑥‖‖².(4.11)
Hence, if 𝑀 is sufficient small such that 2𝛾≥3𝑘₁𝑐^11/2+6𝑘₁𝑐^21/2+7𝑘₁(𝑐₂+𝑐)^1/2, 2𝛾≥3𝑘₁𝑐^11/2+7𝑘₁𝑐^21/2+7𝑘₁(𝑐₂+𝑐)^1/2, then it follows from (4.8)–(4.11), we get 𝑑𝑑𝑡‖𝑣‖²+‖‖𝑣_𝑥‖‖²+𝜌‖𝑣‖²+‖‖𝑣_𝑥‖‖²≤0,(4.12) where 𝜌≥0 is suitable constant.
Applying Gronwall’s inequality, we derive that ‖𝑣(𝑡)‖²+‖‖𝑣_𝑥(𝑡)‖‖²≤‖𝑣(0)‖²+‖‖𝑣_𝑥(0)‖‖²𝑒^−𝜌𝑡,forany𝑡≥0.(4.13)
Since 𝑣 is 𝜔-periodic in 𝑡, then for any positive integer 𝑚 we have ‖𝑣(𝑡)‖²+‖‖𝑣_𝑥(𝑡)‖‖²=‖𝑣(𝑡+𝑚𝜔)‖²+‖‖𝑣_𝑥(𝑡+𝑚𝜔)‖‖².(4.14)
Then we can infer that ‖𝑣(𝑡)‖²+‖‖𝑣_𝑥(𝑡)‖‖²≤‖𝑣(0)‖²+‖‖𝑣_𝑥(0)‖‖²𝑒^{−𝜌(𝑡+𝑚𝜔)}.(4.15)
It follows from 𝑣(0)=𝑣_𝑥(0)=0 that 𝑢(𝑡,𝑥)=𝑢(𝑡,𝑥), which completes the proof of Theorem 4.2.