# On the Study of Local Solutions for a Generalized Camassa-Holm Equation

Authors: Meng Wu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The pseudoparabolic regularization technique is employed to study the local well-posedness of strong solutions for a nonlinear dispersive model, which includes the famous Camassa-Holm equation. The local well-posedness is established in the Sobolev spaceHs(R) with s>3/2 via a limiting procedure.


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## Body

## 1. Introduction

In recent years, extensive research has been carried out worldwide to study highly nonlinear equations including the Camassa-Holm (CH) equation and its various generalizations [1–6]. It is shown in [7–9] that the inverse spectral or scattering approach is a powerful technique to handle the Camassa-Holm equation and analyze its dynamics. It is pointed out in [10–12] that the CH equation gives rise to geodesic flow of a certain invariant metric on the Bott-Virasoro group, and this geometric illustration leads to a proof that the Least Action Principle holds. Li and Olver [13] established the local well-posedness to the CH model in the Sobolev space Hs(R) with s>3/2 and gave conditions on the initial data that lead to finite time blow-up of certain solutions. Constantin and Escher [14] proved that the blow-up occurs in the form of breaking waves, namely, the solution remains bounded but its slope becomes unbounded in finite time. Hakkaev and Kirchev [15] investigated a generalized form of the Camassa-Holm equation with high order nonlinear terms and obtained the orbit stability of the traveling wave solutions under certain assumptions. Lai and Wu [16] discussed a generalized Camassa-Holm model and acquired its local existence and uniqueness. Recently, Li et al. [17] investigated the generalized Camassa-Holm equation
(1.1)ut-utxx+kumux+(m+3)um+1ux=(m+2)umuxuxx+um+1uxxx,
where m≥0 is a natural number and k≥0. The authors in [17] assume that the initial value satisfies the sign condition and establish the global existence of solutions for (1.1).In this paper, we will study the following generalization of (1.1):
(1.2)ut-utxx+kumux+(m+3)um+1ux=(m+2)umuxuxx+um+1uxxx+λ(u-uxx),
where m≥0 is a natural number, k≥0, and λ is a constant.The objective of this paper is to study the local well-posedness of (1.2). Its local well-posedness of strong solutions in the Sobolev space Hs(R) with s>3/2 is investigated by using the pseudoparabolic regularization method. Comparing with the work by Li et al. [17], (1.2) considered in this paper possesses a conservation law different to that in [17] (see Lemma 3.2 in Section 3). Also (1.2) contains a dissipative term λ(u-uxx), which causes difficulty to establish its local and global existence in the Sobolev space. It should be mentioned that the existence and uniqueness of local strong solutions for the generalized nonlinear Camassa-Holm models like (1.2) have never been investigated in the literatures.The organization of this work is as follows. The main result is given in Section2. Section 3 establishes several lemmas, and the last section gives the proof of the main result.


## 2. Main Result

Firstly, we introduce several notations.L
p
=
L
p
(
R
)
(
1
≤
p
<
+
∞
) is the space of all measurable functions h such that ∥h∥Lpp=∫R‍|h(t,x)|pdx<∞. We define L∞=L∞(R) with the standard norm ∥h∥L∞=infm(e)=0supx∈R∖e|h(t,x)|. For any real number s, Hs=Hs(R) denotes the Sobolev space with the norm defined by
(2.1)‖h‖Hs=(∫R(1+|ξ|2)s|h^(t,ξ)|2dξ)1/2<∞,
where  h^(t,ξ)=∫R‍e-ixξh(t,x)dx.ForT>0 and nonnegative number s, C([0,T);Hs(R)) denotes the Frechet space of all continuous Hs-valued functions on [0,T). We set Λ=(1-∂x2)1/2. For simplicity, throughout this paper, we let c denote any positive constant that is independent of parameter ɛ.We consider the Cauchy problem of (1.2)
(2.2)ut-utxx=-km+1(um+1)x-m+3m+2(um+2)x+1m+2∂x3(um+2)-(m+1)∂x(umux2)+umuxuxx+λ(u-uxx),k≥0,m≥0,u(0,x)=u0(x).Now, we give our main results for problem of (2.2).Theorem 2.1.
Suppose that the initial functionu0(x) belongs to the Sobolev space Hs(R) with s>3/2 and λ is a constant. Then, there is a T>0, which depends on ∥u0∥Hs, such that problem (2.2) has a unique solution u(t,x) satisfying
(2.3)u(t,x)∈C([0,T);Hs(R))⋂‍C1([0,T);Hs-1(R)).


## 3. Local Well-Posedness

In order to prove Theorem2.1, we consider the associated regularized problem
(3.1)ut-utxx+ɛutxxxx=-km+1(um+1)x-m+3m+2(um+2)x+1m+2∂x3(um+2)-(m+1)∂x(umux2)+umuxuxx+λ(u-uxx),u(0,x)=u0(x),
where the parameter ɛ  satisfies 0<ɛ<1/4.Lemma 3.1.
Fors≥1 and f(x)∈Hs(R) and letting k1>0  be an integer such that  k1≤s-1,  f,f′,…,fk1 are uniformly continuous bounded functions that converge to 0 at x=±∞.The proof of Lemma3.1 was stated on page 559 by Bona and Smith [18].Lemma 3.2.
Ifu(t,x)∈Hs(s>7/2) is a solution to problem (3.1), it holds that
(3.2)∫R(u2+ux2+ɛuxx2)dx=∫R(u02+u0x2+ɛu0xx2)dx+2λ∫0t∫R(u2+ux2)dx.Proof.
Using Lemma3.1, we have u(t,±∞)=ux(t,±∞)=uxx(t,±∞)=uxxx(t,±∞)=0. The integration by parts results in
(3.3)∫Rum+2uxxxdx=∫Rum+2duxx=um+2uxx∣-∞+∞-(m+2)∫Rum+1uxuxxdx=-(m+2)∫Rum+1uxuxxdx.
Direct calculation and integration by parts give rise to(3.4)12ddt∫R(u2+ux2+ɛuxx2)dx=∫R(uut+uxutx+ɛuxxutxx)dx=∫Ru(ut-utxx+ɛutxxxx)dx=∫Ru[-km+1(um+1)x-m+3m+2(um+2)x+1m+2∂x3(um+2)-(m+1)∂x(umux2)+umuxuxx+λ(u-uxx)km+1]dx=∫Ru[(m+2)umuxuxx+um+1uxxx]dx+λ∫R(u2-uuxx)dx=∫R[(m+2)um+1uxuxx+um+2uxxx]dx+λ∫R(u2+ux2)dx=λ∫R(u2+ux2)dx,
in which we have used (3.3). From (3.4), we obtain the conservation law (3.2).Lemma 3.3.
Lets≥7/2. The function u(t,x) is a solution of problem (3.1) and the initial value u0(x)∈Hs. Then, the following inequality holds:
(3.5)‖u‖H12≤∫R(u02+u0x2+ɛu0xx2)dx,ifλ≤0,‖u‖H12≤e2λt∫R(u02+u0x2+ɛu0xx2)dx,ifλ>0.
For  q∈(0,s-1], there is a constant c independent of  ɛ  such that
(3.6)∫R(Λq+1u)2dx≤∫R[(Λq+1u0)2+ɛ(Λqu0xx)2]dx+c∫0t‖u‖Hq+12(|λ|+(‖u‖L∞m-1+‖u‖L∞m)‖ux‖L∞+‖u‖L∞m-1‖ux‖L∞2)dτ.
For  q∈[0,s-1],  there is a constant  c  independent of  ɛ  such that
(3.7)(1-2ɛ)‖ut‖Hq≤c‖u‖Hq+1(|λ|+(‖u‖L∞m-1+‖u‖L∞m)‖u‖H1+‖u‖L∞m‖ux‖L∞+‖u‖L∞m-1‖ux‖L∞2).The proof of this lemma is similar to that of Lemma  3.5 in [17]. Here we omit it.Lemma 3.4.
Letr  and  q  be real numbers such that  -r<q≤r.  Then,
(3.8)‖uv‖Hq≤c‖u‖Hr‖v‖Hq,ifr>12,‖uv‖Hr+q-1/2≤c‖u‖Hr‖v‖Hq,ifr<12.This lemma can be found in [19] or [20].Lemma 3.5.
Letu0(x)∈Hs(R) with s>3/2. Then, the Cauchy problem (3.1) has a unique solution u(t,x)∈C([0,T];Hs(R)), where T>0 depends on ∥u0∥Hs(R). If s≥7/2, the solution  u∈C([0,+∞);Hs) exists for all time.Proof.
LettingD=(1-∂x2+ɛ∂x4)-1, we know that D:Hs→Hs+4 is a bounded linear operator. Applying the operator D on both sides of the first equation of system (3.1) and then integrating the resultant equation with respect to t over the interval (0,t), we get
(3.9)u(t,x)=u0(x)+∫0tD[-km+1(um+1)x-m+3m+2(um+2)x+1m+2∂x3(um+2)-(m+1)∂x(umux2)+umuxuxx+λ(u-uxx)km+1]dt.
Suppose that both u and v are in the closed ball BM0(0) of radius M0  about the zero function in C([0,T];Hs(R)) and A is the operator in the right-hand side of (3.9). For any fixed t∈[0,T], we obtain
(3.10)‖∫0tD[-km+1(um+1)x-m+3m+2(um+2)x+1m+2∂x3(um+2)-(m+1)∂x(umux2)+umuxuxx+λ(u-uxx)km+1]dt-∫0tD[-km+1(vm+1)x-m+3m+2(vm+2)x+1m+2∂x3(vm+2)-(m+1)∂x(vmvx2)+vmvxvxx+λ(v-vxx)km+1]dt∫0t‖Hs≤TC1(sup⁡0≤t≤T‖u-v‖Hs+sup⁡0≤t≤T‖um+1-vm+1‖Hs+sup⁡0≤t≤T‖um+2-vm+2‖Hs+sup⁡0≤t≤T‖D∂x[umux2-vmvx2]‖Hs+sup⁡0≤t≤T‖D[umuxuxx-vmvxvxx]‖Hs),
where C1 may depend on ɛ. The algebraic property of  Hs0(R)  with  s0>1/2  derives
(3.11)‖um+2-vm+2‖Hs=‖(u-v)(um+1+umv+⋯+uvm+vm+1)‖Hs≤‖(u-v)‖Hs∑j=0m+1‍‖u‖Hsm+1-j‖v‖Hsj,≤M0m+1‖u-v‖Hs,‖um+1-vm+1‖Hs≤M0m‖u-v‖Hs,‖D∂x(umux2-vmvx2)‖Hs≤‖D∂x[um(ux2-vx2)]‖Hs+‖D∂x[vx2(um-vm)]‖Hs≤C(‖um(ux2-vx2)‖Hs-1+‖vx2(um-vm)‖Hs-1)≤CM0m+1‖u-v‖Hs.
Using the first inequality of Lemma 3.4 gives rise to
(3.12)‖D[umuxuxx-vmvxvxx]‖Hs=‖12D[um(ux2)x-vm(vx2)x]‖Hs≤12(‖D[um(ux2-vx2)x]‖Hs+‖D[(vx2)x(um-vm)]‖Hs)≤C(‖um(ux2-vx2)x‖Hs-2+‖(vx2)x(um-vm)‖Hs-2)≤C(‖um‖Hs‖ux2-vx2‖Hs-1+‖vx2‖Hs-1‖um-vm‖Hs)≤CM0m+1‖u-v‖Hs,
where  C  may depend on  ɛ.  From (3.11)-(3.12), we obtain
(3.13)‖Au-Av‖Hs≤θ‖u-v‖Hs,
where θ=TC2(M0m+M0m+1) and C2 is independent of  0<t<T. Choosing T sufficiently small such that  θ<1, we know that A  is a contraction. Similarly, it follows from (3.10) that
(3.14)‖Au‖Hs≤‖u0‖Hs+θ‖u‖Hs.
Choosing T sufficiently small such that  θM0+∥u0∥Hs<M0,  we deduce that A maps BM0(0) to itself. It follows from the contraction-mapping principle that the mapping A has a unique fixed point u in BM0(0). It completes the proof.From the above and Lemma3.2, we have
(3.15)∫R(u2+ux2+ɛuxx2)dx≤e2|λ|t∫R(u02+u0x2+ɛu0xx2)dx.
Therefore,
(3.16)‖ux‖L∞≤Cɛe2|λ|t∫R(u02+u0x2+ɛu0xx2)dx,
which together with Lemma 3.3 completes the proof of the global existence.Settingϕɛ(x)=ɛ-1/4ϕ(ɛ-1/4x) with 0<ɛ<1/4 and uɛ0=ϕɛ⋆u0, we know that uɛ0∈C∞ for any u0∈Hs,s>0. From Lemma 3.5, it derives that the Cauchy problem
(3.17)ut-utxx+ɛutxxxx=-km+1(um+1)x-m+3m+2(um+2)x+1m+2∂x3(um+2)-(m+1)∂x(umux2)+umuxuxx+λ(u-uxx),u(0,x)=uɛ0(x),x∈R,
has a unique solution uɛ(t,x)∈C∞([0,∞);H∞).Furthermore, we have the following.Lemma 3.6.
Fors>0,u0∈Hs, it holds that
(3.18)‖uɛ0x‖L∞≤c‖u0x‖L∞,(3.19)‖uɛ0‖Hq≤c,ifq≤s,(3.20)‖uɛ0‖Hq≤cɛ(s-q)/4,ifq>s,(3.21)‖uɛ0-u0‖Hq≤cɛ(s-q)/4,ifq≤s,(3.22)‖uɛ0-u0‖Hs=o(1),
where c is a constant independent of ɛ.The proof of Lemma3.6 can be found in [16].Remark 3.7.
Fors≥1, using ∥uɛ∥L∞≤c∥uɛ∥H1/2+≤c∥uɛ∥H1,  ∥uɛ∥H12≤c∫R(uɛ2+uɛx2)dx, (3.5), (3.19), and (3.20), we know that,
(3.23)‖uɛ‖L∞2≤c‖uɛ‖H12≤ce2|λ|t∫R(uɛ02+uɛ0x2+ɛuɛ0xx2)dx≤ce2|λ|t(‖uɛ0‖H12+ɛ‖uɛ0‖H22)≤ce|2λ|t(c+cɛ×ɛ(s-2)/2)≤c0e2|λ|t,
where c0 is independent of ɛ and t.Lemma 3.8.
Supposeu0(x)∈Hs(R) with s≥1 such that ∥u0x∥L∞<∞. Let uɛ0 be defined as in system (3.17). Then, there exist two positive constants T and c, which are independent of ɛ, such that the solution uɛ of problem (3.17) satisfies ∥uɛx∥L∞≤c for any t∈[0,T).Here we omit the proof of Lemma3.8 since it is similar to Lemma  3.9 presented in [17].Lemma 3.9 (see Li and Olver [13]).
Ifu  and  f  are functions in Hq+1∩{∥ux∥L∞<∞}, then
(3.24)|∫RΛquΛq(uf)xdx|≤{cq‖f‖Hq+1‖u‖Hq2,q∈(12,1],cq(‖f‖Hq+1‖u‖Hq‖u‖L∞+‖fx‖L∞‖u‖Hq2+‖f‖Hq‖u‖Hq‖ux‖L∞),cq‖f‖Hq+1‖u‖Hq2,q∈(0,∞).Lemma 3.10 (see Lai and Wu [16]).
Foru,v∈Hs(R) with s>3/2,  w=u-v,q>1/2, and a natural number n, it holds that
(3.25)|∫RΛswΛs(un+1-vn+1)xdx|≤c(‖w‖Hs‖w‖Hq‖v‖Hs+1+‖w‖Hs2).Lemma 3.11 (see Lai and Wu [16]).
If1/2<q<min{1,s-1} and s>3/2, then for any functions w,f defined on R, it holds that
(3.26)|∫RΛqwΛq-2(wf)xdx|≤c‖w‖Hq2‖f‖Hq,(3.27)|∫RΛqwΛq-2(wxfx)xdx|≤c‖w‖Hq2‖f‖Hs.Lemma 3.12.
For problem (3.17), s>3/2, and u0∈Hs(R), there exist two positive constants c and M, which are independent of ɛ, such that the following inequalities hold for any sufficiently small ɛ and t∈[0,T):
(3.28)‖uɛ‖Hs≤Mect,‖uɛ‖Hs+k1≤ɛ-k1/4Mect,k1>0,‖uɛt‖Hs+k1≤ɛ-(k1+1)/4Mect,k1>-1.Slightly modifying the methods presented in [16] can complete the proof of Lemma 3.12.Our next step is to demonstrate thatuɛ is a Cauchy sequence. Let uɛ and uδ be solutions of problem (3.17), corresponding to the parameters ɛ and δ, respectively, with 0<ɛ<δ<1/4, and let w=uɛ-uδ. Then, w satisfies the problem
(3.29)(1-ɛ)wt-ɛwxxt+(δ-ɛ)(uδt+uδxxt)=(1-∂x2)-1[-ɛwt+(δ-ɛ)uδt-km+1∂x(uɛm+1-uδm+1)-∂x(uɛm+2-uδm+2)-∂x[∂x(uɛm+1)∂xw+∂x(uɛm+1-uɛm+1)∂xuδ]+[uɛmuɛxuɛxx-uδmuδxuδxx]km+1]-1m+2∂x(uɛm+2-uδm+2)+λw,(3.30)w(x,0)=w0(x)=uɛ0(x)-uδ0(x).Lemma 3.13.
Fors>3/2, u0∈Hs(R), there exists T>0 such that the solution uɛ of (3.17) is a Cauchy sequence in C([0,T];Hs(R))⋂‍C1([0,T];Hs-1(R)).Proof.
Forq with 1/2<q<min{1,s-1}, multiplying both sides of  (3.29) by ΛqwΛq and then integrating with respect to x give rise to
(3.31)12ddt∫R[(1-ɛ)(Λqw)2+ɛ(Λqwx)2]dx=(ɛ-δ)∫R(Λqw)[(Λquδt)+(Λquδxxt)]dx-ɛ∫RΛqwΛq-2wtdx+(δ-ɛ)∫RΛqwΛq-2uδtdx-1m+2∫R(Λqw)Λq(uɛm+2-uδm+2)xdx-km+1∫RΛqwΛq-2(uɛm+1-uδm+1)xdx-∫RΛqwΛq-2(uɛm+2-uδm+2)xdx-∫RΛqwΛq-2[∂x(uɛm+1)∂xw]xdx-∫RΛqwΛq-2[∂x(uɛm+1-uδm+1)∂xuδ]xdx+∫RΛqwΛq-2[uɛmuɛxuɛxx-uδmuδxuδxx]dx+λ∫RΛqwΛqwdx.
It follows from the Schwarz inequality that
(3.32)ddt∫[(1-ɛ)(Λqw)2+ɛ(Λqwx)2]dx≤c{∫R‖Λqw‖L2[(δ-ɛ)(‖Λquδt‖L2+‖Λquδxxt‖L2)+ɛ‖Λq-2wt‖L2+(δ-ɛ)‖Λq-2uδt‖L2]+|λ|∫R(Λqw)2dx+|∫RΛqwΛq(uɛm+2-uδm+2)xdx||∫ΛqwΛq-2(uɛm+1-uδm+1)xdx|+|∫ΛqwΛq-2(uɛm+2-uδm+2)xdx|+|∫RΛqwΛq-2[∂x(uɛm+1)∂xw]xdx|+|∫RΛqwΛq-2[∂x(uɛm+1-uδm+1)∂xuδ]xdx|+|∫RΛqwΛq-2[uɛmuɛxuɛxx-uδmuδxuδxx]dx|}.
Using the first inequality in Lemma 3.9, we have
(3.33)|∫RΛqwΛq(uɛm+2-uδm+2)xdx|=|∫RΛqwΛq(wgm+1)xdx|≤c‖w‖Hq2‖gm+1‖Hq+1,
where  gm+1=∑j=0m+1uɛm+1-juδj. For the last three terms in (3.32), using Lemmas 3.4 and 3.12,  1/2<q<min{1,s-1}, s>3/2, the algebra property of Hs0 with s0>1/2,  and (3.23), we have
(3.34)|∫RΛqwΛq-2(∂x(uɛm+1)∂xw)xdx|≤c‖w‖Hq2‖uɛ‖Hsm+1,(3.35)|∫RΛqwΛq-2(∂x(uɛm+1-uδm+1)∂xuδ)xdx|≤c‖w‖Hq‖uδ‖Hs‖uɛm+1-uδm+1‖Hq≤c‖w‖Hq2‖uδ‖Hs,(3.36)|∫RΛqwΛq-2[uɛmuɛxuɛxx-uδmuδxuδxx]dx|≤c‖w‖Hq‖(uɛm-uδm)(uɛx2)x+uδm[uɛx2-uδx2]x‖Hq-2≤c‖w‖Hq(‖(uɛm-uδm)(uɛx2)x‖Hq-1+‖uδm[uɛx2-uδx2]x‖Hq-2)≤c‖w‖Hq(‖uɛm-uδm‖Hq‖(uɛx2)x‖Hq-1+‖uδm‖Hs‖[uɛx2-uδx2]x‖Hq-2)≤c‖w‖Hq(‖w‖Hq‖gm-1‖Hq‖u‖Hs2+‖uδm‖Hs‖uɛx+uδx‖Hq‖w‖Hq)≤c‖w‖Hq2(‖gm-1‖Hq‖u‖Hs2+‖uδm‖Hs‖uɛx+uδx‖Hq).
Using (3.26), we derive that the inequality
(3.37)|∫RΛqwΛq-2(uɛm+2-uδm+2)xdx|=|∫RΛqwΛq-2(wgm+1)xdx|≤c‖gm+1‖Hq‖w‖Hq2
holds for some constant c, where  gm+1=∑j=0m+1uɛm+1-juδj.  Using the algebra property of Hq with q>1/2, q+1<s and Lemma 3.11, we have ∥gm∥Hq+1≤c for t∈(0,T~]. Then, it follows from (3.28) and (3.33)–(3.37) that there is a constant c depending on T~ such that the estimate
(3.38)ddt∫R[(1-ɛ)(Λqw)2+ɛ(Λqwx)2]dx≤c(δγ‖w‖Hq+‖w‖Hq2)
holds for any t∈[0,T~), where γ=1 if s≥3+q  and  γ=(1+s-q)/4 if s<3+q. Integrating (3.38) with respect to t, one obtains the estimate
(3.39)12‖w‖Hq2=12∫R(Λqw)2dx≤∫R[(1-ɛ)(Λqw)2+ɛ(Λqw)2]dx≤∫R[(Λqw0)2+ɛ(Λqw0x)2]dx+c∫0t(δγ‖w‖Hq+‖w‖Hq2)dτ.
Applying the Gronwall inequality and using (3.20) and (3.22) yield
(3.40)‖u‖Hq≤cδ(s-q)/4ect+δγ(ect-1)
for any t∈[0,T~).
Multiplying both sides of (3.29) by ΛswΛs and integrating the resultant equation with respect to x, one obtains
(3.41)12ddt∫R[(1-ɛ)(Λsw)2+ɛ(Λswx)2]dx=(ɛ-δ)∫R(Λsw)[(Λsuδt)+(Λsuδxxt)]dx-ɛ∫RΛswΛs-2wtdx+(δ-ɛ)∫RΛswΛs-2uδtdx-km+1∫R(Λsw)Λs(uɛm+1-uδm+1)xdx-1m+2∫R(Λsw)Λs(uɛm+2-uδm+2)xdx-∫RΛswΛs-2(uɛm+2-uδm+2)xdx-∫RΛswΛs-2[∂x(uɛm+1)∂xw]xdx-∫RΛswΛs-2[∂x(uɛm+1-uδm+1)∂xuδ]xdx+∫RΛswΛs-2[uɛmuɛxuɛxx-uδmuδxuδxx]dx+λ∫R(Λsw)2dx.
From Lemma 3.12, we have
(3.42)|∫RΛswΛs-2(uɛm+2-uδm+2)xdx|≤c3‖gm+1‖Hs‖w‖Hs2.
From Lemma 3.10, it holds that
(3.43)|∫RΛswΛs(uɛm+2-uδm+2)xdx|≤c(‖w‖Hs‖w‖Hq‖uδ‖Hs+1+‖w‖Hs2).
Using the Cauchy-Schwartz inequality and the algebra property of Hs0 with s0>1/2, for s>3/2, we have
(3.44)|∫RΛswΛs-2[∂x(uɛm+1)∂xw]xdx|=|∫RΛqwΛs-2[∂x(uɛm+1)∂xw]xdx|≤c‖Λsw‖L2‖Λs-2[∂x(uɛm+1)∂xw]x‖L2≤c‖w‖Hq‖∂x(uɛm+1)∂xw‖Hs-1≤c‖uɛHsm+1‖‖w‖Hs2,(3.45)|∫RΛswΛs-2[∂x(uɛm+1-uδm+1)∂xuδ]xdx|≤c‖w‖Hs‖Λs-2[∂x(uɛm+1-uδm+1)∂xuδ]x‖L2≤c‖uδ‖Hs‖gm‖Hs‖w‖Hs2,(3.46)|∫RΛswΛs-2[uɛmuɛxuɛxx-uδmuδxuδxx]dx|≤c‖w‖Hs(‖(uɛm-uδm)(uɛx2)x‖Hs-2+‖uδm[uɛx2-uδx2]x‖Hs-2)≤c‖w‖Hs(‖uɛm-uδm‖Hs‖(uɛx2)x‖Hs-2+‖uδm‖Hs‖[uɛx2-uδx2]x‖Hs-2)≤c‖w‖Hs2,
in which we have used Lemma 3.4 and the bounded property of ∥uɛ∥Hs and ∥uδ∥Hs (see Remark 3.7). It follows from (3.41)–(3.46) and the inequalities (3.28) and (3.40) that there exists a constant c depending on m such that
(3.47)ddt∫R[(1-ɛ)(Λsw)2+ɛ(Λswx)2]dx≤2δ(‖uδt‖Hs+‖uδxxt‖Hs+‖Λs-2wt‖L2+‖Λs-2uδt‖)‖w‖Hs+c(‖w‖Hs2+‖w‖Hq‖w‖Hs‖uδ‖Hs+1)≤c(δγ1‖w‖Hs+‖w‖Hs2),
where γ1=min(1/4,(s-q-1)/4)>0. Integrating (3.47) with respect to t leads to the estimate
(3.48)12‖w‖Hs2≤∫R[(1-ɛ)(Λsw)2+ɛ(Λswx)2]dx≤∫R[(Λsw0)2+ɛ(Λsw0x)2]dx+c∫0t(δγ1‖w‖Hs+‖w‖Hs2)dτ.
It follows from the Gronwall inequality and (3.48) that
(3.49)‖w‖Hs≤(2∫R[(Λsw0)2+ɛ(Λsw0x)2]dx)1/2ect+δγ1(ect-1)≤c1(‖w0‖Hs+δ3/4)ect+δγ1(ect-1),
where c1 is independent of ɛ and δ.
Then, (3.22) and the above inequality show that
(3.50)‖w‖Hs⟶0asɛ⟶0,δ⟶0.
Next, we consider the convergence of the sequence {uɛt}. Multiplying both sides of (3.29) by Λs-1wtΛs-1 and integrating the resultant equation with respect to x, we obtain
(3.51)(1-ɛ)‖wt‖Hs-12+1m+2∫R(Λs-1wt)Λs-1(uɛm+2-uδm+2)xdx+∫R[-ɛ(Λs-1wt)(Λs-1wxxt)+(δ-ɛ)(Λs-1wt)Λs-1(uδt+uδxxt)]dx=∫R(Λs-1wt)Λs-3[-ɛwt+(δ-ɛ)uδt-km+1∂x(uɛm+1-uδm+1)-∂x(uɛm+2-uδm+2)∫R(Λs-1wt)Λs-3-∂x[∂x(uɛm+1)∂xw+∂x(uɛm+1-uɛm+1)∂xuδ]∫R(Λs-1wt)Λs-3+[uɛmuɛxuɛxx-uδmuδxuδxx]km+1]dx+λ∫R‍Λs-1wtΛs-1wdx.
It follows from inequalities (3.28) and the Schwartz inequality that there is a constant c depending on T~ and m such that
(3.52)(1-ɛ)‖wt‖Hs-12≤c(δ1/2+‖w‖Hs+‖w‖s-1)‖wt‖Hs-1+ɛ‖wt‖Hs-12
Hence
(3.53)12‖wt‖Hs-12≤(1-2ɛ)‖wt‖Hs-12≤c(δ1/2+‖w‖Hs+‖w‖Hs-1)‖wt‖Hs-1,
which results in
(3.54)12‖wt‖Hs-1≤c(δ1/2+‖w‖Hs+‖w‖Hs-1).
It follows from (3.40) and (3.50) that wt→0 as ɛ, δ→0 in the Hs-1 norm. This implies that uɛ is a Cauchy sequence in the spaces C([0,T);Hs(R)) and C([0,T);Hs-1(R)), respectively. The proof is completed.


## 4. Proof of the Main Result

We consider the problem(4.1)(1-ɛ)ut-ɛutxx=(1-∂x2)-1[-km+1(um+1)x-m+3m+2(um+2)x+1m+2∂x3(um+2)-(m+1)∂x(umux2)+umuxuxxkm+1]+λu,u(0,x)=uɛ0(x).
Letting u(t,x) be the limit of the sequence uɛ and taking the limit in problem (4.1) as ɛ→0, from Lemma 3.13, we know that u is a solution of the problem
(4.2)ut=(1-∂x2)-1[-km+1(um+1)x-m+3m+2(um+2)x+1m+2∂x3(um+2)-(m+1)∂x(umux2)+umuxuxxkm+1]+λu,u(0,x)=u0(x),
and hence u is a solution of problem (4.2) in the sense of distribution. In particular, if s≥4, u is also a classical solution. Let u and v be two solutions of (4.2) corresponding to the same initial value u0 such that u, v∈C([0,T);Hs(R)). Then, w=u-v satisfies the Cauchy problem
(4.3)wt=(1-∂x2)-1{∂x[-km+1wgm-m+3m+2wgm+1+1m+2∂x2(wgm+1)-∂x(um+1)∂xw-∂x(um+1-vm+1)∂xvkm+1]+umuxuxx-vmvxvxxkm+1}+λw,w(0,x)=0.
For any 1/2<q<min{1,s-1}, applying the operator ΛqwΛq to both sides of (4.3) and integrating the resultant equation with respect to x, we obtain the equality
(4.4)12ddt‖w‖Hq2=∫R(Λqw)Λq-2{∂x[-km+1wgm-m+3m+2wgm+1+1m+2∂x2(wgm+1)12ddt‖w‖Hq2=∫R(Λqw)Λq-2-∂x(um+1)∂xw-∂x(um+1-vm+1)∂xvkm+1]+umuxuxx12ddt‖w‖Hq2=∫R-vmvxvxxkm+1}dx+|λ|‖w‖Hq2.
By the similar estimates presented in Lemma 3.13, we have
(4.5)ddt‖w‖Hq2≤c~‖w‖Hq2.
Using the Gronwall inequality leads to the conclusion that
(4.6)‖w‖Hq≤0×ec~t=0
for t∈[0,T~). This completes the proof.



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*Source: 164876-2012-08-28.xml*