Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 164876, 17 pages
doi:10.1155/2012/164876
Research Article
On the Study of Local Solutions for
a Generalized Camassa-Holm Equation
Meng Wu
School of Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China
Correspondence should be addressed to Meng Wu, wumeng@swufe.edu.cn
Received 23 May 2012; Revised 30 June 2012; Accepted 18 July 2012
Academic Editor: Yong Hong Wu
Copyright q 2012 Meng Wu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
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 space H s R with s > 3/2 via
a limiting procedure.
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 BottVirasoro 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 H s 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
ut − utxx kum ux m 3um1 ux m 2um ux uxx um1 uxxx ,
1.1
2
Abstract and Applied Analysis
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:
ut − utxx kum ux m 3um1 ux m 2um ux uxx um1 uxxx λu − uxx ,
1.2
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 wellposedness of strong solutions in the Sobolev space H s 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 λu − 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 Section 2.
Section 3 establishes several lemmas, and the last section gives the proof of the main result.
2. Main Result
Firstly, we introduce several notations.
Lp Lp R 1 ≤ p < ∞ is the space of all measurable functions h such that
p
hLp R |ht, x|p dx < ∞. We define L∞ L∞ R with the standard norm hL∞ infme0 supx∈R\e |ht, x|. For any real number s, H s H s R denotes the Sobolev space with
the norm defined by
hH s 2 1/2
s 1 |ξ| ht, ξ dξ
< ∞,
2
2.1
R
ξ e−ixξ ht, xdx.
where ht,
R
For T > 0 and nonnegative number s, C0, T ; H s R denotes the Frechet space of all
1/2
continuous H s -valued functions on 0, T . We set Λ 1 − ∂2x . For simplicity, throughout
this paper, we let c denote any positive constant that is independent of parameter ε.
We consider the Cauchy problem of 1.2
k m1 m 3 m2 1
u
u
∂3x um2
−
x
x
m1
m2
m2
− m 1∂x um u2x um ux uxx λu − uxx , k ≥ 0, m ≥ 0,
ut − utxx −
u0, x u0 x.
Now, we give our main results for problem of 2.2.
2.2
Abstract and Applied Analysis
3
Theorem 2.1. Suppose that the initial function u0 x belongs to the Sobolev space H s R with s >
3/2 and λ is a constant. Then, there is a T > 0, which depends on u0 H s , such that problem 2.2 has
a unique solution ut, x satisfying
ut, x ∈ C0, T ; H s R
C1 0, T ; H s−1 R .
2.3
3. Local Well-Posedness
In order to prove Theorem 2.1, we consider the associated regularized problem
k m1 m 3 m2 1
u
u
∂3x um2
−
x
x
m1
m2
m2
− m 1∂x um u2x um ux uxx λu − uxx ,
ut − utxx εutxxxx −
3.1
u0, x u0 x,
where the parameter ε satisfies 0 < ε < 1/4.
Lemma 3.1. For s ≥ 1 and fx ∈ H s R and letting k1 > 0 be an integer such that k1 ≤ s − 1,
f, f , . . . , f k1 are uniformly continuous bounded functions that converge to 0 at x ±∞.
The proof of Lemma 3.1 was stated on page 559 by Bona and Smith 18.
Lemma 3.2. If ut, x ∈ H s s > 7/2 is a solution to problem 3.1, it holds that
t u2 u2x εu2xx dx u20 u20x εu20xx dx 2λ
u2 u2x dx.
R
0
R
3.2
R
Proof. Using Lemma 3.1, we have ut, ±∞ ux t, ±∞ uxx t, ±∞ uxxx t, ±∞ 0. The
integration by parts results in
u
R
m2
uxxx dx u
m2
duxx u
R
m2
uxx |∞
−∞
−m 2
u
R
m1
ux uxx dx.
− m 2
um1 ux uxx dx
R
3.3
4
Abstract and Applied Analysis
Direct calculation and integration by parts give rise to
1 d
2 dt
R
u2 u2x εu2xx dx
uut ux utx εuxx utxx dx
R
uut − utxx εutxxxx dx
R
k m1 m 3 m2 1
u
u
∂3x um2
−
x
x
m1
m2
m2
m 2
m
−m 1∂x u ux u ux uxx λu − uxx dx
u −
R
3.4
m
m1
u2 − uuxx dx
u m 2u ux uxx u uxxx dx λ
R
R
m1
m2
u2 u2x dx
m 2u ux uxx u uxxx dx λ
R
λ
R
u2 u2x dx,
R
in which we have used 3.3. From 3.4, we obtain the conservation law 3.2.
Lemma 3.3. Let s ≥ 7/2. The function ut, x is a solution of problem 3.1 and the initial value
u0 x ∈ H s . Then, the following inequality holds:
u2H 1
≤
u2H 1
≤e
u20 u20x εu20xx dx,
R
2λt
R
u20
u20x
εu20xx
if λ ≤ 0,
3.5
dx,
if λ > 0.
For q ∈ 0, s − 1, there is a constant c independent of ε such that
R
Λ
q1
2
u dx ≤
Λ
R
c
t
0
q1
u0
2
u2H q1
2
εΛ u0xx dx
q
|λ| um−1
L∞
um
L∞
ux L∞ 2
um−1
L∞ ux L∞
3.6
dτ.
For q ∈ 0, s − 1, there is a constant c independent of ε such that
m
m
m−1
2
.
u
1 − 2εut H q ≤ cuH q1 |λ| um−1
u
u
u
u
u
1
∞
∞
∞
∞
∞
∞
x
x
H
L
L
L
L
L
L
3.7
Abstract and Applied Analysis
5
The proof of this lemma is similar to that of Lemma 3.5 in 17. Here we omit it.
Lemma 3.4. Let r and q be real numbers such that −r < q ≤ r. Then,
uvH q ≤ cuH r vH q ,
if r >
uvH rq−1/2 ≤ cuH r vH q ,
1
,
2
3.8
1
if r < .
2
This lemma can be found in 19 or 20.
Lemma 3.5. Let u0 x ∈ H s R with s > 3/2. Then, the Cauchy problem 3.1 has a unique
solution ut, x ∈ C0, T ; H s R, where T > 0 depends on u0 H s R . If s ≥ 7/2, the solution
u ∈ C0, ∞; H s exists for all time.
−1
Proof. Letting D 1 − ∂2x ε∂4x , we know that D : H s → H s4 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
ut, x u0 x t
0
k m1 m 3 m2 1
u
u
∂3x um2
−
x
x
m1
m2
m2
m 2
m
−m 1∂x u ux u ux uxx λu − uxx dt.
D −
3.9
Suppose that both u and v are in the closed ball BM0 0 of radius M0 about the zero function in
C0, T ; H s R and A is the operator in the right-hand side of 3.9. For any fixed t ∈ 0, T ,
we obtain
t
k m1 m 3 m2 1
u
u
∂3x um2
−
D −
0
x
x
m1
m2
m2
−m 1∂x um u2x um ux uxx λu − uxx dt
−
t
0
D −
k m1 m 3 m2 1
v
v
∂3x vm2
−
x
x
m1
m2
m2
m 2
m
−m 1∂x v vx v vx vxx λv − vxx dt
s
≤ T C1
3.10
H
sup u − vH s sup um1 − vm1 0≤t≤T
Hs
0≤t≤T
sup D∂x um u2x − vm vx2 0≤t≤T
Hs
sup um2 − vm2 Hs
0≤t≤T
sup Du ux uxx − v vx vxx H s
m
0≤t≤T
m
,
6
Abstract and Applied Analysis
where C1 may depend on ε. The algebraic property of H s0 R with s0 > 1/2 derives
m2
− vm2 u
Hs
u − v um1 um v · · · uvm vm1 Hs
≤ u − vH s
m1
m1−j
j
uH s
vH s ,
j0
≤ M0m1 u − vH s ,
m1
− vm1 s ≤ M0m u − vH s ,
u
D∂x um u2x − vm vx2 Hs
H
≤ D∂x um u2x − vx2 Hs
≤ C um u2x − vx2 H s−1
3.11
D∂x vx2 um − vm vx2 um − vm Hs
H s−1
≤ CM0m1 u − vH s .
Using the first inequality of Lemma 3.4 gives rise to
Du ux uxx − v vx vxx H s
m
m
1 m 2 m
D u ux − v vx2
x
x H s
2
1 D um u2x − vx2
s D vx2 um − vm s
x
x
2
H
H
≤ C um u2x − vx2 s−2 vx2 um − vm s−2
≤
x H
≤ C um H s u2x − vx2 H s−1
vx2 x
H
H
um − vm H s
s−1
≤ CM0m1 u − vH s ,
3.12
where C may depend on ε. From 3.11-3.12, we obtain
Au − AvH s ≤ θu − vH s ,
3.13
where θ T C2 M0m M0m1 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
AuH s ≤ u0 H s θuH s .
3.14
Choosing T sufficiently small such that θM0 u0 H s < 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.
Abstract and Applied Analysis
7
From the above and Lemma 3.2, we have
R
u2 u2x εu2xx dx ≤ e2|λ|t
u20 u20x εu20xx dx.
3.15
R
Therefore,
ux L∞ ≤ Cε e2|λ|t
R
u20 u20x εu20xx dx,
3.16
which together with Lemma 3.3 completes the proof of the global existence.
Setting φε x ε−1/4 φε−1/4 x with 0 < ε < 1/4 and uε0 φε u0 , we know that
uε0 ∈ C∞ for any u0 ∈ H s , s > 0. From Lemma 3.5, it derives that the Cauchy problem
k m1 m 3 m2 1
u
u
∂3x um2
−
x
x
m1
m2
m2
− m 1∂x um u2x um ux uxx λu − uxx ,
ut − utxx εutxxxx −
u0, x uε0 x,
3.17
x ∈ R,
has a unique solution uε t, x ∈ C∞ 0, ∞; H ∞ .
Furthermore, we have the following.
Lemma 3.6. For s > 0, u0 ∈ H s , it holds that
uε0x L∞ ≤ cu0x L∞ ,
uε0 H q ≤ c,
if q ≤ s,
uε0 H q ≤ cεs−q/4 ,
uε0 − u0 H q ≤ cεs−q/4 ,
if q > s,
if q ≤ s,
uε0 − u0 H s o1,
where c is a constant independent of ε.
The proof of Lemma 3.6 can be found in 16.
3.18
3.19
3.20
3.21
3.22
8
Abstract and Applied Analysis
Remark 3.7. For s ≥ 1, using uε L∞ ≤ cuε H 1/2 ≤ cuε H 1 , uε 2H 1 ≤ c
3.19, and 3.20, we know that,
uε 2L∞ ≤ cuε 2H 1 ≤ ce2|λ|t
u2ε u2εx dx, 3.5,
R
u2ε0 u2ε0x εu2ε0xx dx
≤ ce2|λ|t uε0 2H 1 εuε0 2H 2
≤ ce
R
|2λ|t
c cε × ε
s−2/2
3.23
≤ c0 e2|λ|t ,
where c0 is independent of ε and t.
Lemma 3.8. Suppose u0 x ∈ H s R with s ≥ 1 such that u0x L∞ < ∞. Let uε0 be defined as in
system 3.17. Then, there exist two positive constants T and c, which are independent of ε, such that
the solution uε of problem 3.17 satisfies uεx L∞ ≤ c for any t ∈ 0, T .
Here we omit the proof of Lemma 3.8 since it is similar to Lemma 3.9 presented in
17.
Lemma 3.9 see Li and Olver 13. If u and f are functions in H q1 ∩ {ux L∞ < ∞}, then
⎧
1
⎪
2
⎪
,1 ,
cq f H q1 uH q , q ∈
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
⎨cq f H q1 uH q uL∞
Λq uΛq uf dx ≤
⎪
x
⎪
R
fx u2 q f q u q ux ⎪
⎪
H
L∞ ,
H
L∞
H
⎪
⎪
⎪
⎪
⎪
⎩
q ∈ 0, ∞.
3.24
Lemma 3.10 see Lai and Wu 16. For u, v ∈ H s R with s > 3/2, w u − v, q > 1/2, and a
natural number n, it holds that
Λs wΛs un1 − vn1 dx ≤ c w s w q v s1 w2 s .
H
H
H
H
x
3.25
R
Lemma 3.11 see Lai and Wu 16. If 1/2 < q < min{1, s−1} and s > 3/2, then for any functions
w, f defined on R, it holds that
Λq wΛq−2 wf dx ≤ cw2 q f q ,
H
H
x
3.26
Λq wΛq−2 wx fx dx ≤ cw2 q f s .
H
H
x
3.27
R
R
Abstract and Applied Analysis
9
Lemma 3.12. For problem 3.17, s > 3/2, and u0 ∈ H s R, there exist two positive constants c and
M, which are independent of ε, such that the following inequalities hold for any sufficiently small ε
and t ∈ 0, T :
uε H s ≤ Mect ,
uε H sk1 ≤ ε−k1 /4 Mect ,
3.28
k1 > 0,
uεt H sk1 ≤ ε−k1 1/4 Mect ,
k1 > −1.
Slightly modifying the methods presented in 16 can complete the proof of
Lemma 3.12.
Our next step is to demonstrate that uε is a Cauchy sequence. Let uε and uδ be solutions
of problem 3.17, corresponding to the parameters ε and δ, respectively, with 0 < ε < δ < 1/4,
and let w uε − uδ . Then, w satisfies the problem
1 − εwt − εwxxt δ − εuδt uδxxt −1 −εwt δ − εuδt −
1 − ∂2x
k
∂x um1
− ∂x um2
− um1
− um2
ε
ε
δ
δ
m1
∂x w ∂x um1
∂x uδ
− um1
− ∂x ∂x um1
ε
ε
ε
m
um
u
u
−
u
u
u
−
εx
εxx
δx
δxx
ε
δ
3.29
1
m2
∂x um2
λw,
−
u
ε
δ
m2
wx, 0 w0 x uε0 x − uδ0 x.
3.30
Lemma 3.13. For s > 3/2, u0 ∈ H s R, there exists T > 0 such that the solution uε of 3.17 is a
Cauchy sequence in C0, T ; H s R C1 0, T ; H s−1 R.
Proof. For q with 1/2 < q < min{1, s − 1}, multiplying both sides of 3.29 by Λq wΛq and then
integrating with respect to x give rise to
1 d
2 dt
1 − εΛq w2 εΛq wx 2 dx
R
ε − δ
Λq wΛq−2 wt dx
Λ wΛ uδt Λ uδxxt dx − ε
q
q
q
R
δ − ε
Λ wΛ
q
R
q−2
1
uδt dx −
m2
R
R
− um2
dx
Λq wΛq um2
ε
δ
x
10
Abstract and Applied Analysis
k
m1
q
q−2
m2
m2
u
Λq wΛq−2 um1
−
u
dx
−
Λ
wΛ
−
u
dx
ε
ε
δ
δ
x
x
m1 R
R
q
q−2
m1
∂x uε
∂x w dx −
∂
Λ wΛ
Λq wΛq−2 ∂x um1
− um1
u
dx
−
x
δ
ε
δ
−
x
R
Λ wΛ
q
q−2
R
um
ε uεx uεxx
−
x
R
um
δ uδx uδxx
dx λ
Λq wΛq wdx.
R
3.31
It follows from the Schwarz inequality that
d 1 − εΛq w2 εΛq wx 2 dx
dt
≤ c Λq wL2 δ − εΛq uδt L2 Λq uδxxt L2 εΛq−2 wt L2
|λ|
R
δ − εΛq−2 uδt 2
L
2
q
q
q
m2
m2
dx
Λ w dx Λ wΛ uε − uδ
x
R
m1
q
q−2
m2
m2
Λq wΛq−2 um1
uε − uδ
− uδ
dx Λ wΛ
dx
ε
x
x
q
q−2
m1
m1
Λq wΛq−2 ∂x um1
∂
∂
u
∂
w
dx
Λ
wΛ
−
u
u
dx
x
x
x δ
ε
ε
δ
x
x
R
R
m
.
Λq wΛq−2 um
u
u
−
u
u
u
dx
εx
εxx
δx
δxx
ε
δ
R
3.32
Using the first inequality in Lemma 3.9, we have
m2
q
q
Λq wΛq um2
−
u
dx
Λ
wΛ
dx
wg
m1
ε
δ
x
x
R
R
≤ cw2H q gm1 H q1 ,
3.33
m1−j j
uδ . For the last three terms in 3.32, using Lemmas 3.4 and 3.12,
where gm1 m1
j0 uε
1/2 < q < min{1, s − 1}, s > 3/2, the algebra property of H s0 with s0 > 1/2, and 3.23,
Abstract and Applied Analysis
11
we have
Λq wΛq−2 ∂x um1
∂x w dx ≤ cw2H q uε m1
ε
Hs ,
x
3.34
R
m1
Λq wΛq−2 ∂x um1
∂
−
u
u
dx
x δ
ε
δ
x
R
≤ cwH q uδ H s um1
− um1
ε
δ 3.35
Hq
≤ cw2H q uδ H s ,
m
Λq wΛq−2 um
ε uεx uεxx − uδ uδx uδxx dx
R
2 m
2
2
≤ cwH q um
uεx um
u
−
u
ε − uδ
εx
δx δ
x H q−2
x
2 m
uεx ≤ cwH q um
ε − uδ
x
H q−1
2
2
um
δ uεx − uδx x
2 m ≤ cwH q um
ε − uδ H q uεx
x H q−1
H q−2
2
2
s
u
um
−
u
εx
δx δ H
3.36
x H q−2
≤ cwH q wH q gm−1 H q u2H s um
δ H s uεx uδx H q wH q
≤ cw2H q gm−1 H q u2H s um
δ H s uεx uδx H q .
Using 3.26, we derive that the inequality
m2
q
q−2
Λq wΛq−2 um2
− uδ
dx Λ wΛ
wgm1 x dx
ε
x
R
R
≤ cgm1 H q w2H q
holds for some constant c, where gm1 m1
3.37
m1−j j
uδ .
Using the algebra property of H q
with q > 1/2, q 1 < s and Lemma 3.11, we have gm H q1 ≤ 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
d
dt
j0
uε
1 − εΛq w2 εΛq wx 2 dx ≤ c δγ wH q w2H q
R
3.38
12
Abstract and Applied Analysis
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
1
1
2
Λq w2 dx
wH q 2
2 R
≤
1 − εΛq w2 εΛq w2 dx
3.39
R
≤
t
δγ wH q w2H q dτ.
Λq w0 2 εΛq w0x 2 dx c
0
R
Applying the Gronwall inequality and using 3.20 and 3.22 yield
uH q ≤ cδs−q/4 ect δγ ect − 1
3.40
for any t ∈ 0, T .
Multiplying both sides of 3.29 by Λs wΛs and integrating the resultant equation with
respect to x, one obtains
1 d
2 dt
1 − εΛs w2 εΛs wx 2 dx
R
ε − δ
Λs wΛs−2 wt dx
Λ wΛ uδt Λ uδxxt dx − ε
s
s
s
R
R
k
m1
−
u
dx
δ − ε Λ wΛ uδt dx −
Λs wΛs um1
ε
δ
x
m1 R
R
3.41
1
s
s
m2
m2
s
s−2
m2
m2
−
u ε − uδ
dx −
Λ wΛ
dx
Λ wΛ uε − uδ
x
x
m2 R
R
s
s−2
m1
m1
−
∂x uε
∂x w dx −
∂
Λ wΛ
Λs wΛs−2 ∂x um1
−
u
u
dx
x
δ
ε
δ
s
s−2
x
R
R
R
m
Λs wΛs−2 um
ε uεx uεxx − uδ uδx uδxx dx λ
x
Λs w2 dx.
R
From Lemma 3.12, we have
m2
Λs wΛs−2 um2
− uδ
dx ≤ c3 gm1 H s w2H s .
ε
x
3.42
R
From Lemma 3.10, it holds that
m2
Λs wΛs um2
− uδ
dx ≤ c wH s wH q uδ H s1 w2H s .
ε
x
R
3.43
Abstract and Applied Analysis
13
Using the Cauchy-Schwartz inequality and the algebra property of H s0 with s0 > 1/2, for
s > 3/2, we have
Λs wΛs−2 ∂x um1
∂x w dx
ε
x
R
Λq wΛs−2 ∂x um1
∂
w
dx
x
ε
x
R
≤ cΛs wL2 Λs−2 ∂x um1
∂x w ε
3.44
x L2
∂
≤ cwH q ∂x um1
w
x
ε
H s−1
2
≤ cuε m1
H s wH s ,
m1
Λs wΛs−2 ∂x um1
∂
−
u
u
dx
x δ
ε
δ
x
R
≤ cwH s Λs−2 ∂x um1
∂x uδ − um1
ε
δ
3.45
x L2
≤ cuδ H s gm H s w2H s ,
m
Λs wΛs−2 um
ε uεx uεxx − uδ uδx uδxx dx
R
2 m
≤ cwH s um
uεx ε − uδ
x H s−2
2
2
u
um
−
u
εx
δx δ
2 m ≤ cwH s um
ε − uδ H s uεx
x H s−2
x H s−2
2
2
um
δ H s uεx − uδx 3.46
x H s−2
≤ cw2H s ,
in which we have used Lemma 3.4 and the bounded property of uε H s and uδ H s 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
d
dt
1 − εΛs w2 εΛs wx 2 dx
R
≤ 2δ uδt H s uδxxt H s Λs−2 wt L2
c w2H s wH q wH s uδ H s1
≤ c δγ1 wH s w2H s ,
Λs−2 uδt wH s
3.47
14
Abstract and Applied Analysis
where γ1 min1/4, s − q − 1/4 > 0. Integrating 3.47 with respect to t leads to the estimate
1
w2H s ≤
2
≤
R
R
1 − εΛs w2 εΛs wx 2 dx
t
δγ1 wH s w2H s dτ.
Λ w0 εΛ w0x dx c
s
2
3.48
2
s
0
It follows from the Gronwall inequality and 3.48 that
wH s
1/2
2
2
s
s
≤ 2
ect δγ1 ect − 1
Λ w0 εΛ w0x dx
R
≤ c1 w0 H s δ
3/4
e δ
ct
γ1
e −1 ,
3.49
ct
where c1 is independent of ε and δ.
Then, 3.22 and the above inequality show that
wH s −→ 0
as ε −→ 0,
δ −→ 0.
3.50
Next, we consider the convergence of the sequence {uεt }. Multiplying both sides of 3.29 by
Λs−1 wt Λs−1 and integrating the resultant equation with respect to x, we obtain
1
Λs−1 wt Λs−1 um2
− um2
dx
1 −
ε
δ
x
m2 R
−ε Λs−1 wt Λs−1 wxxt δ − ε Λs−1 wt Λs−1 uδt uδxxt dx
εwt 2H s−1
R
R
Λs−1 wt Λs−3 −εwt δ − εuδt −
k
m2
m2
∂x um1
−
∂
u
− um1
−
u
x
ε
ε
δ
δ
m1
∂x w ∂x um1
∂x uδ
− ∂x ∂x um1
− um1
ε
ε
ε
m
um
u
u
−
u
u
u
Λs−1 wt Λs−1 wdx.
dx
λ
ε εx εxx
δ δx δxx
R
3.51
It follows from inequalities 3.28 and the Schwartz inequality that there is a constant
c depending on T and m such that
1 − εwt 2H s−1 ≤ c δ1/2 wH s ws−1 wt H s−1 εwt 2H s−1
3.52
Abstract and Applied Analysis
15
Hence
1
wt 2H s−1 ≤ 1 − 2εwt 2H s−1
2
≤ c δ1/2 wH s wH s−1 wt H s−1 ,
3.53
1
wt H s−1 ≤ c δ1/2 wH s wH s−1 .
2
3.54
which results in
It follows from 3.40 and 3.50 that wt → 0 as ε, δ → 0 in the H s−1 norm. This
implies that uε is a Cauchy sequence in the spaces C0, T ; H s R and C0, T ; H s−1 R,
respectively. The proof is completed.
4. Proof of the Main Result
We consider the problem
1 − εut − εutxx 1 −
∂2x
−1 m 3 m2 1
k m1 u
u
∂3x um2
−
x
x
m1
m2
m2
−m 1∂x um u2x um ux uxx λu,
−
4.1
u0, x uε0 x.
Letting ut, x be the limit of the sequence uε and taking the limit in problem 4.1 as ε → 0,
from Lemma 3.13, we know that u is a solution of the problem
−1 −
ut 1 − ∂2x
m 3 m2 1
k m1 u
u
∂3x um2
−
x
x
m1
m2
m2
−m 1∂x um u2x um ux uxx λu,
4.2
u0, 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
16
Abstract and Applied Analysis
initial value u0 such that u, v ∈ C0, T ; H s R. Then, w u−v satisfies the Cauchy problem
−1 ∂x −
wt 1 − ∂2x
m3
1
k
wgm −
wgm1 ∂2x wgm1
m1
m2
m2
−∂x um1 ∂x w − ∂x um1 − vm1 ∂x v um ux uxx − vm vx vxx λw,
w0, x 0.
4.3
For any 1/2 < q < min{1, s − 1}, applying the operator Λq wΛq to both sides of 4.3 and
integrating the resultant equation with respect to x, we obtain the equality
1 d
w2H q 2 dt
q−2
wΛ
∂x −
Λ
q
R
m3
1
k
wgm −
wgm1 ∂2 wgm1
m1
m2
m2 x
m1
m1
m1
−∂x u
∂x w − ∂x u
∂x v um ux uxx
−v
4.4
−vm vx vxx dx |λ|w2H q .
By the similar estimates presented in Lemma 3.13, we have
d
w2H q ≤ cw2H q .
dt
4.5
Using the Gronwall inequality leads to the conclusion that
wH q ≤ 0 × ect 0
4.6
for t ∈ 0, T . This completes the proof.
Acknowledgments
Thanks are given to the referees whose suggestions were very helpful in improving the paper.
This work is supported by the Fundamental Research Funds for the Central Universities
JBK120504.
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