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

Authors: Chunyu Shen
Journal: International Journal of Differential Equations
 (2011)

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## 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.


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## Body

## 1. Introduction

The Camassa-Holm equation(1.1)ut-utxx+3uux=2uxuxx+uuxxx,t>0,x∈R,
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∈Hs(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,(1.2)‖u(t,⋅)‖L∞≤22‖u(t,⋅)‖H1≤22‖u0(⋅)‖H1.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 is(1.3)ut-utxx+3uux-2uxuxx-uuxxx+L(u)=f(t,x),t>0,x∈R,
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.Withf=0 and L(u)=γ(1-∂x2)u, (1.3) becomes weakly dissipative Camassa-Holm equation(1.4)ut-utxx+3uux+γ(u-uxx)=2uxuxx+uuxxx,t>0,x∈R,
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-∂x2)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 laws(1.5)I1=∫Sudx,I2=∫S(u2+ux2)dx,
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 equation(1.6)ut-utxx+3uux-2uxuxx-uuxxx+γ(u-uxx)=f(t,x),t>0,x∈R,(1.7)u(t,x+L)=u(t,x),t>0,x∈R,(1.8)u(t+ω,x)=u(t,x),t>0,x∈R,
where γ(1-∂x2)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 Section2, 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.LetX be a Banach space, we denote by Ck(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 Ck(ω;X) is ∥u∥Ck(ω;X)=sup0≤t≤ω{∑i=0k∥Dtiu∥X}.For1≤p≤∞, the space Lp(ω;X) is the set of ω-periodic X-valued measurable functions on ℝ such that
(2.1)‖u‖Lp(ω;X)={(∫0ω‖u‖Xpdt)1/p<∞,1≤p<∞sup⁡0≤t≤ω‖u‖X<∞,p=∞.The spaceWk,p(ω;X) denote the set of functions which belong to Lp(ω;X) together with their derivatives up to order k, and we write Wk,2(ω;X)=Hk(ω;X) in particular when X is a Hilbert space.Lp(Ω) and Hm(Ω) are classical Sobolev spaces. For simplicity, we write ∥·∥Lp(Ω) by ∥·∥p as p≠2 and ∥·∥L2(Ω) by ∥·∥.The following inequalities (see [47]) will be used in the proofs later(2.2)‖u‖∞≤k1‖u‖H1.(2.3)‖Dju‖p≤k2‖u‖Hmθ‖u‖1-θ,
where Dju=(∂ju)/(∂xj), 1/p=j+θ(1/2-m)+(1-θ)(1/2) as 0≤j<m,  j/m≤θ≤1.(2.4)‖u‖≤k3‖ux‖,∫Ωu(x)dx=0.


## 3. A Priori Estimates

In this section, we first prove that (1.6)–(1.8) have a sequence of approximate solutions {un}n=1∞, then give a prior, estimates about {un}n=1∞.We denote the unbounded linear operatorAu=-uxx on X=L2∩{u∣u(x+L)=u(x),∫Ωudx=0}, then the set of all linearly independent eigenvectors {wj}j=0∞ of A, that is, Awj=λjwj, with 0<λ1≤λ2≤⋯≤λj→∞, is an orthonormal basis of L2(Ω). For any n and a group of function {ajn(t)}j=1n, where ajn(t)(j=1,2,…,n)∈C1(ω;ℝ), the function un=∑j=1najn(t)wj∈C1(ω;Hn) is called an approximate solution to (1.6)–(1.8) if it satisfies the equation as follows:(3.1)(unt-unxxt+γ(un-unxx),wj)=(Nun+f,wj),j=1,…,n,
where Nun=-3ununx+2unxunxx+ununxxx and Hn=span{w1,w2,…,wn}. By the classical theory of ordinary differential equations, for any fixed vn(t)=∑j=1nbjn(t)wj∈C1(ω;Hn), the equation (unt-unxxt+γ(un-unxx),wj)=(Nvn+f,wj), j=1,…,n has a unique ω-periodic solution un and the mapping F:vn→un is continuous and compact in C1(ω;Hn). Hence by Leray-Schauder fixed point theorem, we want to prove the existence of an approximate solution only to show sup0≤t≤ω∥un∥2≤c for all possible solution of (3.1) replaced by λNun(0≤λ≤1) instead of nonlinear term Nun, where c is a constant which only depends on L, ε, ω, γ, and f.Lemma 3.1.
Iff∈C1(ω;H-1(Ω)), then
(3.2)sup⁡0≤t≤ω(‖un‖2+‖unx‖2)≤c1,
where c1 is a constant which only depends on L, ω, ε, γ, k3, and f, M=sup0≤t≤ω{∥f(t,x)∥H-1(Ω)2} and d1=min{2γ,2γ-ε}>0.Proof.
Multiplying (3.1) by ajn(t) and summing up over j from 1 to n, we obtain
(3.3)(unt-unxxt+γ(un-unxx),un)=(Nun+f,un).
Then, we can get(3.4)12ddt(‖un‖2+‖unx‖2)+γ(‖un‖2+‖unx‖2)=(Nun+f,un).
Notice that-3∫Ωun2unxdx=0, 2∫Ωununxunxxdx+∫Ωun2unxxxdx=0.
From Young’s inequality, we have∫Ωfundx≤(ε/2)∥unx∥2+k32M/2ε, where ε>0 is a constant.
According to the above relations, we can derive from (3.4) that
(3.5)ddt(‖un‖2+‖unx‖2)+d1(‖un‖2+‖unx‖2)≤k32Mε,
where d1=min{2γ,2γ-ε}>0.
Considering the time periodicity ofun and integrating (3.5) over [0,ω], we get
(3.6)d1∫0ω(‖un‖2+‖unx‖2)dt≤ωk32Mε.
Hence, there existst*∈[0,ω) such that ∥un(t*)∥2+∥unx(t*)∥2≤k32M/d1ε.
From (3.5), we have (d/dt)(∥un∥2+∥unx∥2)≤k32M/ε.
Integrating the above inequality with respect tot from t* to t∈[t*,t*+ω], we deduce that
(3.7)‖un(t)‖2+‖unx(t)‖2≤‖un(t*)‖2+‖unx(t*)‖2+ωk32Mε≤(1d1+ω)k32Mε.
Hence, we infer(3.8)sup⁡0≤t≤ω(‖un‖2+‖unx‖2)≤(1d1+ω)k32Mε≜c1,
which concludes our proof.From Lemma3.1 and Leray-Schauder fixed point theorem, (3.1) has solution {un}n=1∞, which is also a sequence of approximate solutions of (1.6)–(1.8). In order to obtain the convergence of sequence {un}n=1∞, we need to give a priori estimates for the high-order derivers of {un}n=1∞.Lemma 3.2.
Iff∈C1(ω;H-1(Ω)), then
(3.9)sup⁡0≤t≤ω(‖unx‖2+‖unxx‖2)≤c2,
where c2 is a constant which only depends on L, ω, ε, γ, λn, k1, k2, k3, and f, M=sup0≤t≤ω{∥f(t,x)∥H-1(Ω)2} and d2=min{2γ-(13/2)ελn,2γ-(21/2)ε}>0.Proof.
Multiplying (3.1) by -λjajn(t) and summing up over j from 1 to n, we have
(3.10)(unt-unxxt+γ(un-unxx),unxx)=(Nun+f,unxx).
The above equation yields(3.11)-12ddt(‖unx‖2+‖unxx‖2)-γ(‖unx‖2+‖unxx‖2)=(Nun+f,unxx).
From Young’s inequality, we have(3.12)|∫Ωfunxxdx|≤ε‖unxxx‖2+k32M4ε,
where ε>0 is a constant.
From (2.2), (3.8), and Young’s inequality, we can deduce that
(3.13)|∫Ωununxunxxdx|≤‖un‖∞∫Ω|unxunxx|dx≤k1‖un‖H1∫Ω|unxunxx|dx≤k1c11/2(εk1c11/2‖unxx‖2+k1c11/24ε‖unx‖2)≤ε‖unxx‖2+k12c124ε.
From (2.3), (3.8), Cauchy-Schwarz inequality, Young’s inequality, and Lemma 3.1, we get
(3.14)|∫Ωununxxunxxxdx|=|-12∫Ωunxunxx2dx|≤12‖unx‖‖unxx‖42≤12c11/2k22‖un‖1/2‖un‖H33/2≤34ε‖un‖H32+c13k2864ε3≤34ε‖unxx‖2+34ε‖unxxx‖2+34εc1+c13k2864ε3,(3.15)‖unxxx‖2=∫Ω|(∑j=1najn(t)wj)xxx|2dx=∫Ω|(∑j=1nλjajn(t)wj)x|2dx≤λn‖unx‖2.
Taking (3.11)–(3.15) into account, we can infer that
(3.16)ddt(‖unx‖2+‖unxx‖2)+d2(‖unx‖2+‖unxx‖2)≤k32M2ε+3k12c122ε+9εc12+3c13k2832ε3,
where d2=min{2γ-(13/2)ελn,2γ-(21/2)ε}>0.
Integrating (3.16) about t from 0 to ω and considering the time periodicity of un, we get
(3.17)d2∫0ω(‖unx‖2+‖unxx‖2)dt≤(k32M2ε+3k12c122ε+9εc12+3c13k2832ε3)ω.
Hence, there existst*∈[0,ω) such that
(3.18)‖unx(t*)‖2+‖unxx(t*)‖2≤1d2(k32M2ε+3k12c122ε+9εc12+3c13k2832ε3).
From (3.16), we have
(3.19)‖unx(t)‖2+‖unxx(t)‖2≤‖unx(t*)‖2+‖unxx(t*)‖2+(k32M2ε+3k12c122ε+9εc12+3c13k2832ε3)ω.
Then we can obtain(3.20)sup⁡0≤t≤ω(‖unx‖2+‖unxx‖2)≤(1d2+ω)(k32M2ε+3k12c122ε+9εc12+3c13k2832ε3)≜c2,
which concludes our proof.In the following, we continue to establish a priori estimates for high-order derivers of the approximate solution{un}n=1∞ by an inductive argument.Lemma 3.3.
For anyk≥0, if f∈C1(ω;Hk-1(Ω)), then
(3.21)sup⁡0≤t≤ω(‖Dk+1un‖2+‖Dk+2un‖2)≤c,
where c is a constant which only depends on L,  ω,  ε,  γ,  λn,  k,  k1,  k2,  k3,  f and d3={2γ-16ελn,2γ-14ε}>0.Proof.
By Lemma3.2, we know the conclusion of Lemma 3.3 holds for k=0.
Assume that fork≤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+1λjm+1ajn(t) and summing up over j from 1 to n, we have
(3.22)(-1)m+112ddt(‖Dm+1un‖2+‖Dm+2un‖2)+(-1)m+1γ(‖Dm+1un‖2+‖Dm+2un‖2)=(Nun+f,D2(m+1)un).
Follow the same methods discussed in Lemma3.2, we have
(3.23)|∫ΩfD2(m+1)undx|=|∫ΩDm-1fDm+3undx|≤ε‖Dm+3un‖2+14ε‖Dm-1f‖2.
From the conclusion of Lemma3.3 for k≤m-1, (2.2), (2.4) and Young’s inequality, we can deduce that
(3.24)|∫ΩununxD2(m+1)undx|=|∫Ω(∑i=0m+1Cm+1iDiunDm+1-iunx)Dm+1undx|≤∫Ω|unDm+2unDm+1un|dx+∫Ω|(∑i=1m+1Cm+1iDiunDm+1-iunx)Dm+1un|dx≤ε‖Dm+2un‖2+c(ε,k1,k3)‖Dm+1un‖2+c(m,k1)≤ε‖Dm+2un‖2+c(ε,k1,k3,m).
Similarly, we can also deduce that(3.25)|∫ΩununxxxD2(m+1)undx|=|∫Ω(∑i=0mCmiDiunDm-iunxxx)Dm+2undx|≤∫Ω|unDm+3unDm+2un|dx+∫Ω|Cm1Dun(Dm+2un)2|dx+∫Ω|Cm2D2unDm+1unDm+2un|dx+∫Ω|(∑i=3mCmiDiunDm+3-iun)Dm+2un|dx≤‖un‖∞∫Ω|Dm+3unDm+2un|dx+m‖Dun‖∞∫Ω|Dm+2un|2dx+Cm2‖D2un‖∞∫Ω|Dm+1unDm+2un|dx+∑i=3mCmi‖Diun‖∞‖Dm+3-iun‖∞∫Ω|Dm+2un|dx≤c(k1,k3,m)(∫Ω|Dm+3unDm+2un|dx+∫Ω|Dm+2un|2dx+∫Ω|Dm+1unDm+2un|dx+∫Ω|Dm+2un|dx).
From the conclusion of Lemma3.3 for k≤m-1, Young’s inequality and (2.3), we have
(3.26)c(k1,k3,m)∫Ω|Dm+3unDm+2un|dx≤ε‖Dm+3un‖2+c(k1,k3,m,ε)‖Dm+2un‖2≤ε‖Dm+3un‖2+c(k1,k2,k3,m,ε)×‖un‖Hm+32(m+2)/m+3‖un‖2(1-(m+2)/(m+3))≤ε‖Dm+3un‖2+ε‖un‖Hm+32+c(k1,k2,k3,m,ε)‖un‖2≤2ε‖Dm+3un‖2+ε‖Dm+2un‖2+c(k1,k2,k3,m,ε),c(k1,k3,m)∫Ω|(Dm+2un)2|dx=c(k1,k3,m)‖Dm+2un‖2≤c(k1,k2,k3,m)‖un‖Hm+32(m+2)/(m+3)‖un‖2(1-(m+2)/(m+3))≤ε‖Dm+3un‖2+ε‖Dm+2un‖2+c(k1,k2,k3,m,ε),c(k1,k3,m)∫Ω|Dm+1unDm+2un|dx≤ε‖Dm+2un‖2+c(k1,k3,m,ε)‖Dm+1un‖2≤ε‖Dm+2un‖2+c(k1,k3,m,ε),c(k1,k3,m)∫Ω|Dm+2un|dx≤ε‖Dm+2un‖2+c(k1,k3,m,ε,L).
Combining (3.25) and the above inequality, we can get
(3.27)|∫ΩununxxxD2(m+1)undx|≤3ε‖Dm+3un‖2+4ε‖Dm+2un‖2+c(k1,k2,k3,m,ε,L).
Similarly,(3.28)|∫ΩunxunxxD2(m+1)undx|≤∫Ω|unxDm+1unDm+3un|dx+∫Ω|(∑i=1m-1Cm-1iDi+1unDm+1-iun)Dm+3un|dx≤‖unx‖∞∫Ω|Dm+1unDm+3un|dx+∑i=1m-1Cm-1i‖Di+1un‖∞‖Dm+1-iun‖∞∫Ω|Dm+3un|dx≤2ε‖Dm+3un‖2+c(m,k1,ε,L).
Taking (3.22)–(3.24) and (3.27)-(3.28) into account, we can deduce that
(3.29)ddt(‖Dm+1un‖2+‖Dm+2un‖2)+2γ(‖Dm+1un‖2+‖Dm+2un‖2)≤16ε‖Dm+3un‖2+14ε‖Dm+2un‖2+c(k1,k2,k3,m,ε,f,L).
From the above relation, we can infer(3.30)ddt(‖Dm+1un‖2+‖Dm+2un‖2)+d3(‖Dm+1un‖2+‖Dm+2un‖2)≤c(k1,k2,k3,m,ε,f,L),
where d3={2γ-16ελn,2γ-14ε}>0.
Integrating (3.30) about t from 0 to ω, there exists t*∈[0,ω) such that
(3.31)‖Dm+1un(t*)‖2+‖Dm+2un(t*)‖2≤c(k1,k2,k3,m,ε,f,L)d3.
From (3.30), we have
(3.32)ddt(‖Dm+1un‖2+‖Dm+2un‖2)≤c(k1,k2,k3,m,ε,f,L).
Integrating the above inequality fromt* to t∈[t*,t*+ω] and with (3.31), we can easily obtain
(3.33)sup⁡0≤t≤ω(‖Dm+1un‖2+‖Dm+2un‖2)≤(1d3+ω)c(k1,k2,k3,m,ε,f,L)≜c.
The proof is completed.Lemma 3.4.
For anyk≥0, if f∈C1(ω;Hk+1(Ω)), then
(3.34)sup⁡0≤t≤ω(‖Dkunt‖2+‖Dk+1unt‖2)≤c
where c is a constant which only depends on L, ω, ε, γ, λn, k, k1, k2, k3, and f.Proof.
We first prove the conclusion of Lemma3.4 holds for k=0. Multiplying (3.1) by ajn′(t) and summing up over j from 1 to n, we have
(3.35)‖unt‖2+‖unxt‖2=(Nun+f-γ(un-unxx),unt).
By Lemma3.3, if f∈C1(ω;H1(Ω)), then we have ∥un∥H42≤c. Hence,
(3.36)|(Nun+f-γ(un-unxx),unt)|≤‖Nun+f-γ(un-unxx)‖‖unt‖≤c‖unt‖.
Therefore, from (3.35) and (3.36), it is easy to know that
(3.37)sup⁡0≤t≤ω(‖unt‖2+‖unxt‖2)≤c.
Assume that the conclusion of Lemma3.4 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+1λjm+1ajn′(t) and summing up over j from 1 to n, we have
(3.38)(-1)m+1(‖Dm+1unt‖2+‖Dm+2unt‖2)=(Nun+f-γ(un-unxx),D2(m+1)unt).
By Lemma3.3, if f∈C1(ω;Hm+2(Ω)), then ∥Dkun∥2≤c for k≤m+5. Hence,
(3.39)|(Nun+f-γ(un-unxx),D2(m+1)unt)|≤‖Dm+1[Nun+f-γ(un-unxx)]‖‖Dm+1unt‖≤c‖Dm+1unt‖.
Taking (3.38) and (3.39) into account, it follows
(3.40)sup⁡0≤t≤ω(‖Dm+1unt‖2+‖Dm+2unt‖2)≤c.
This completes the proof of Lemma3.4 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 {un}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 Lemmas3.1–3.4 and standard compactness arguments, we conclude that there is a subsequence which we denote also by {un} such that for any K≥0, if f∈C1(ω;Hk+1(Ω)), we have(4.1)un(t)⟶u(t),weakly*inL∞(ω;Hk+4(Ω)),un(t)⟶u(t),stronglyinL∞(ω;Hk+3(Ω)),unt(t)⟶ut(t),weakly*inL∞(ω;Hk+1(Ω)),unt(t)⟶ut(t),stronglyinL∞(ω;Hk(Ω)).From the above lemmas, we know that the nonlinear terms are well defined(4.2)‖ununx-uux‖≤‖un(unx-ux)‖+‖ux(un-u)‖≤‖un‖∞‖unx-ux‖+‖ux‖∞‖un-u‖⟶0,
as n→∞, uniformly in t,(4.3)‖unxunxx-uxuxx‖≤‖unx(unxx-uxx)‖+‖uxx(unx-ux)‖≤‖unx‖∞‖unxx-uxx‖+‖uxx‖∞‖unx-ux‖⟶0,
as n→∞, uniformly in t,(4.4)‖ununxxx-uuxxx‖≤‖un(unxxx-uxxx)‖+‖uxxx(un-u)‖≤‖un‖∞‖unxxx-uxxx‖+‖un-u‖∞‖uxxx‖≤‖un‖∞‖unxxx-uxxx‖+k1‖un-u‖H1‖uxxx‖⟶0,
as n→∞, uniformly in t.Consequently, it follows that(4.5)(ut-uxxt+γ(u-uxx),η)=(Nu+f,η),η∈Lper2.Thanks to the estimates obtained in the previous section, we have(4.6)ut-uxxt+γ(u-uxx)=Nu+f,
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.
Givenf∈C1(ω;Hk+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;Hk+4(Ω))∩W1,∞(ω;Hk(Ω)).Under the assumption of Theorem4.1, 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 Theorem4.1 holds. If M is sufficiently small, then the time periodic solution of (1.6)–(1.8) in Theorem 4.1 is unique.Proof.
Letu and u̅ be any two time periodic solutions of (1.6)–(1.8). With v=u-u̅, we can get from (1.6) that
(4.7)vt-vxxt+γ(v-vxx)=Nu-Nu̅.
Taking the inner product of (4.7) with v, we have
(4.8)12ddt(‖v‖2+‖vx‖2)+γ(‖v‖2+‖vx‖2)=(Nu-Nu̅,v).
Since,(4.9)|(-3uux+3u̅u̅x,v)|≤|(-3uvx,v)|+|(-3u̅xv,v)|≤32‖u‖∞(‖v‖2+‖vx‖2)+3‖u̅x‖∞‖v‖2≤(3k12c11/2+3k1c21/2)‖v‖2+3k12c11/2‖vx‖2,(4.10)|(2uxuxx-2u̅xu̅xx,v)|≤|(2uxvxx,v)|+|(2u̅xxvx,v)|≤‖uxx‖∞(‖v‖2+‖vx‖2)+2‖ux‖∞‖vx‖2+‖u̅xx‖∞(‖v‖2+‖vx‖2)≤2k1(c2+c)1/2‖v‖2+[2k1(c2+c)1/2+2k1c21/2]‖vx‖2,(4.11)|(uuxxx-u̅u̅xxx,v)|≤|(uvxxx,v)|+|(u̅xxxv,v)|≤∫Ω|uxxvvx|dx+32∫Ω|uxvx2|dx+2∫Ω|u̅xxvvx|dx≤12‖uxx‖∞(‖v‖2+‖vx‖2)+32‖ux‖∞‖vx‖2+‖u̅xx‖∞(‖v‖2+‖vx‖2)≤3k12(c2+c)1/2‖v‖2+[3k12(c2+c)1/2+3k12c21/2]‖vx‖2.
Hence, ifM is sufficient small such that 2γ≥3k1c11/2+6k1c21/2+7k1(c2+c)1/2, 2γ≥3k1c11/2+7k1c21/2+7k1(c2+c)1/2, then it follows from (4.8)–(4.11), we get
(4.12)ddt(‖v‖2+‖vx‖2)+ρ(‖v‖2+‖vx‖2)≤0,
where ρ≥0 is suitable constant.
Applying Gronwall’s inequality, we derive that(4.13)(‖v(t)‖2+‖vx(t)‖2)≤(‖v(0)‖2+‖vx(0)‖2)e-ρt,foranyt≥0.
Sincev is ω-periodic in t, then for any positive integer m we have
(4.14)‖v(t)‖2+‖vx(t)‖2=‖v(t+mω)‖2+‖vx(t+mω)‖2.
Then we can infer that(4.15)(‖v(t)‖2+‖vx(t)‖2)≤(‖v(0)‖2+‖vx(0)‖2)e-ρ(t+mω).
It follows fromv(0)=vx(0)=0 that u(t,x)=u̅(t,x), which completes the proof of Theorem 4.2.



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*Source: 463416-2011-10-26.xml*