Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 158126, 14 pages
doi:10.1155/2012/158126
Research Article
The Local Strong and Weak Solutions for
a Generalized Novikov Equation
Meng Wu and Yue Zhong
Department of Applied Mathematics, Southwestern University of Finance and Economics,
Chengdu 610074, China
Correspondence should be addressed to Meng Wu, wumeng@swufe.edu.cn
Received 8 December 2011; Accepted 13 January 2012
Academic Editor: Shaoyong Lai
Copyright q 2012 M. Wu and Y. Zhong. 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 Kato theorem for abstract differential equations is applied to establish the local well-posedness
of the strong solution for a nonlinear generalized Novikov equation in space C0, T , H s R ∩
C1 0, T , H s−1 R with s > 3/2. The existence of weak solutions for the equation in lower-order
Sobolev space H s R with 1 ≤ s ≤ 3/2 is acquired.
1. Introduction
The Novikov equation with cubic nonlinearities takes the form
ut − utxx 4u2 ux 3uux uxx u2 uxxx ,
1.1
which was derived by Vladimir Novikov in a symmetry classification of nonlocal partial
differential equations 1. Using the perturbed symmetry approach, Novikov was able to
isolate 1.1 and investigate its symmetries. A scalar Lax pair for it was discovered in 1, 2
and was shown to be related to a negative flow in the Sawada-Kotera hierarchy. Many
conserved quantities were found as well as a bi-Hamiltonian structure. The scattering theory
was employed by Hone et al. 3 to find nonsmooth explicit soliton solutions with multiple
peaks for 1.1. This multiple peak property is common with the Camassa-Holm and
Degasperis-Procesi equations see 4–10. Ni and Zhou 11 proved that the Novikov equation associated with initial value is locally well-posedness in Sobolev space H s with s > 3/2
by using the abstract Kato theorem. Two results about the persistence properties of the strong
solution for 1.1 were established. It is shown in 12 that the local well-posedness for the
periodic Cauchy problem of the Novikov equation in Sobolev space H s with s > 5/2.
2
Abstract and Applied Analysis
The orbit invariants are employed to get the existence of periodic global strong solution if
the Sobolev index s ≥ 3 and a sign condition holds. For analytic initial data, the existence and
uniqueness of analytic solutions for 1.1 are also obtained in 12.
In this paper, motivated by the work in 7, 13, we study the following generalized
Novikov equation:
ut − utxx 4u2 ux 3uux uxx u2 uxxx β∂x ux N ,
1.2
where N ≥ 1 is a natural number. In fact, 1.1 has the property
R
u2 u2x dx constant.
1.3
Due to the term β∂x ux N appearing in 1.2, the conservation law 1.3 for 1.2 is
not valid. This brings us a difficulty to obtain the bounded estimates for the solution of 1.2.
However, we will overcome this difficulty to investigate the local existence and uniqueness
of the solution to 1.2 subject to initial value u0 x ∈ H s R with s > 3/2. Meanwhile, a
sufficient condition is presented to guarantee the existence of local weak solution for 1.2.
The main tasks of this work are two-fold. Firstly, by using the Kato theorem for
abstract differential equations, we establish the local existence and uniqueness of solutions
for the 1.2 with any β and arbitrary positive integer N in space C0, T , H s R C1 0, T ,
H s−1 R with s > 3/2. Secondly, it is shown that there exist local weak solutions in lowerorder Sobolev space H s R with 1 ≤ s ≤ 3/2. The ideas of proving the second result come
from those presented in Li and Olver 8.
2. Main Results
Firstly, some notations are presented as follows.
The space of all infinitely differentiable functions φt, x with compact support in
0, ∞ × R is denotedby C0∞ . Lp Lp R1 ≤ p < ∞ is the space of all measurable functions
p
h such that 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 .
The Cauchy problem for 1.2 is written in the form
ut − utxx −
4 3
1
u
∂3x u3 − 2∂x uu2x uux uxx β∂x ux N ,
x
3
3
u0, x u0 x,
2.2
Abstract and Applied Analysis
3
which is equivalent to
1
3 ut u2 ux Λ−2 −3u2 ux − ∂x uu2x − u3x β∂x ux N ,
2
2
2.3
u0, x u0 x.
Now, we state our main results.
Theorem 2.1. Let u0 x ∈ H s R with s > 3/2. Then problem 2.2 or problem 2.3 has a unique
solution ut, x ∈ C0, T ; H s R C1 0, T ; H s−1 R, where T > 0 depends on u0 H s R .
Theorem 2.2. Suppose that u0 x ∈ H s with 1 ≤ s ≤ 3/2 and u0x L∞ < ∞. Then there exists a
T > 0 such that 1.2 subject to initial value u0 x has a weak solution ut, x ∈ L2 0, T , H s in the
sense of distribution and ux ∈ L∞ 0, T × R.
3. Local Well-Posedness
Consider the abstract quasilinear evolution equation
dv
Avv fv,
dt
t ≥ 0,
v0 v0 .
3.1
Let X and Y be Hilbert spaces such that Y is continuously and densely embedded in X, and
let Q : Y → X be a topological isomorphism. Let LY, X be the space of all bounded linear
operators from Y to X. If X Y , we denote this space by LX. We state the following conditions in which ρ1 , ρ2 , ρ3 , and ρ4 are constants depending on max{yY , zY }.
I Ay ∈ LY, X for y ∈ X with
A y − Az w ≤ ρ1 y − z w ,
Y
X
X
y, z, w ∈ Y,
3.2
and Ay ∈ GX, 1, β i.e., Ay is quasi-m-accretive, uniformly on bounded sets
in Y .
II QAyQ−1 Ay By, where By ∈ LX is bounded, uniformly on bounded
sets in Y . Moreover,
B y − Bz w ≤ ρ2 y − z w ,
X
Y
X
y, z ∈ Y, w ∈ X.
3.3
III f : Y → Y extends to a map from X into X, is bounded on bounded sets in Y , and
satisfies
f y − fz ≤ ρ3 y − z ,
Y
Y
f y − fz ≤ ρ4 y − z ,
X
X
y, z ∈ Y,
y, z ∈ Y.
3.4
4
Abstract and Applied Analysis
Kato Theorem (see [14])
Assume that I, II, and III hold. If v0 ∈ Y , there is a maximal T > 0 depending only on
v0 Y and a unique solution v to problem 3.1 such that
v v·, v0 ∈ C0, T ; Y 3.5
C1 0, T ; X.
Moreover, the map v0 → v·, v0 is a continuous map from Y to the space
C0, T ; Y 3.6
C1 0, T ; X.
For problem 2.3, we set Au u2 ∂x , Y H s R, X H s−1 R, Λ 1 − ∂2x 1/2
1
3 fu Λ−2 −3u2 ux − ∂x uu2x − u3x β∂x ux N ,
2
2
,
3.7
and Q Λ. In order to prove Theorem 2.1, we only need to check that Au and fu satisfy
assumptions I–III.
Lemma 3.1 Ni and Zhou 11. The operator Au u2 ∂x with u ∈ H s R, s > 3/2 belongs to
GH s−1 , 1, β.
Lemma 3.2 Ni and Zhou 11. Let Au u2 ∂x with u ∈ H s and s > 3/2. Then Au ∈
LH s , H s−1 for all u ∈ H s . Moreover,
Au − AzwH s−1 ≤ ρ1 u − zH s−1 wH s ,
u, z, w ∈ H s R.
3.8
Lemma 3.3 Ni and Zhou 11. For s > 3/2, u, z ∈ H s and w ∈ H s−1 , it holds that Bu Λ, u2 ∂x Λ−1 ∈ LH s−1 for u ∈ H s and
Bu − BzwH s−1 ≤ ρ2 u − zH s wH s−1 .
3.9
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 ,
uvH r1 ≤ cuL∞ vH r1 ,
1
,
2
if r <
1
,
2
3.10
if r1 ≤ 0.
The above first two inequalities of this lemma can be found in 14, 15, and the third
inequality can be found in 7.
Abstract and Applied Analysis
5
Lemma 3.5. Letting u, z ∈ H s with s > 3/2, then fu is bounded on bounded sets in H s and
satisfies
fu − fz s ≤ ρ3 u − zH s ,
H
fu − fz s−1 ≤ ρ4 u − zH s−1 .
H
3.11
3.12
Proof. Using the algebra property of the space H s0 with s0 > 1/2 and s − 1 > 1/2, we have
−2
Λ ∂x uu2x − ∂x zz2x s
H
2
2
≤ cuux − zzx s−1
H
2
≤ cu − zH s−1 ux s−1 zH s−1 u2x − z2x s−1
H
≤ cu −
H
zH s u2H s
zH s−1 u − zH s−1 u zH s−1
≤ cu − zH s u2H s zH s−1 uH s zH s ,
−2
Λ ∂x ux N − zx N s
H
N
N
≤ cux − zx s−1
3.13
H
≤ cux − zx H s−1
N−1
j0
N−j
j
ux H s−1 zx H s−1 ≤ cu − zH s
It follows from Lemma 3.4 and s − 1 > 1/2 that
−2
Λ ∂x uu2x − ∂x zz2x s−1
H
2
≤ cuux − zz2x s−2
H
≤ cu − zu2x s−2 z u2x − z2x s−2
H
H
2
2
≤ cu − zH s−2 ux s−1 zH s−1 ux − z2x H
N−1
N−j
j
uH s zH s .
j0
H s−2
≤ cu − zH s−1 ux 2H s−1 zH s−1 ux − zx H s−2 ux zx H s−1
≤ cu − zH s−1 u2H s zH s−1 uH s zH s ,
−2
Λ ∂x ux N − zx N s−1
H
N−1
N−j j N
N
≤ cux − zx s−2 ≤ cux − zx H s−2 u
z
x
x
H
j0
s−1
H
≤ cu − zH s−1
N−1
j0
N−j
j
ux H s−1 zx H s−1 ≤ cu − zH s−1
N−1
N−j
j
uH s zH s .
j0
3.14
6
Abstract and Applied Analysis
From 3.7, and 3.13, we know that 3.11 is valid, while inequality 3.12 follows from
3.14.
Proof of Theorem 2.1. Using the Kato Theorem, Lemmas 3.1–3.3 and 3.5, we know that system
2.2 or problem 2.3 has a unique solution
ut, x ∈ C0, T ; H s R
C1 0, T ; H s−1 R .
3.15
4. Existence of Weak Solutions
For s ≥ 2, using the first equation of system 1.3 derives
d
dt
u 2
R
u2x
2β
t
0
dx 0,
uN1
dτ
x
4.1
from which we have the conservation law
u 2
R
u2x
2β
t
0
uN1
dτ
x
dx Lemma 4.1 Kato and Ponce 15. If r ≥ 0, then H r
u20 u20x dx.
R
4.2
L∞ is an algebra. Moreover
uvr ≤ cuL∞ vr ur vL∞ ,
4.3
where c is a constant depending only on r.
Lemma 4.2 Kato and Ponce 15. Letting r > 0. If u ∈ H r
Λr , uvL2 ≤ c ∂x uL∞ Λr−1 v
L2
W 1,∞ and v ∈ H r−1
Λr uL2 vL∞ .
L∞ , then
4.4
Lemma 4.3. Let s ≥ 3/2 and the function ut, x is a solution of problem 2.2 and the initial data
u0 x ∈ H s . Then the following results hold:
uL∞ ≤ uH 1 ≤ cu0 H 1 e|β|
t
0
ux N−1
L∞ dτ
4.5
.
For q ∈ 0, s − 1, there is a constant c such that
2 2
Λq1 u dx ≤
Λq1 u0
dx
R
R
c
t
0
u2H q1
ux L∞ uL∞ ux 2L∞
ux N−1
L∞
4.6
dτ.
Abstract and Applied Analysis
7
For q ∈ 0, s − 1, there is a constant c such that
.
ut H q ≤ cuH q1 uL∞ uH 1 uL∞ ux L∞ ux 2L∞ ux N−1
L∞
4.7
Proof. The identity u2H 1 R u2 u2x dx, 4.2, and the Gronwall inequality result in 4.5.
Using ∂2x −Λ2 1 and the Parseval equality gives rise to
Λq1 u Λq1 u2 ux dx 3 Λq uΛq u2 ux dx.
Λq uΛq ∂3x u3 dx −3
4.8
R
R
R
For q ∈ 0, s − 1, applying Λq uΛq to both sides of the first equation of system 2.2
and integrating with respect to x by parts, we have the identity
1 q 2
Λ u Λq ux 2 dx
2 R
−3 Λq uΛq u2 ux dx
R
4.9
Λq1 u Λq1 u2 ux dx 2 Λq ux Λq uu2x dx
−
R
R
q
q
q
Λ uΛ uux uxx dx − β Λ ux Λq ux N dx.
R
R
We will estimate the terms on the right-hand side of 4.9 separately. For the first term, by
using the Cauchy-Schwartz inequality and Lemmas 4.1 and 4.2, we have
Λq uΛq u2 ux dx
R
Λq u Λq u2 ux − u2 Λq ux dx Λq uu2 Λq ux dx
4.10
R
R
≤ cuH q 2uL∞ ux L∞ uH q ux L∞ uL∞ uH q uL∞ ux L∞ Λq u2L2
≤ cu2H q uL∞ ux L∞ .
Using the above estimate to the second term yields
q1
q1
2
Λ
u
dx
≤ cu2H q1 uL∞ ux L∞ .
u
Λ
u
x
4.11
R
For the third term, using the Cauchy-Schwartz inequality and Lemma 4.2, we obtain
Λq ux Λq uu2x dx
R
≤ Λq ux L2 Λq uu2x 2 ≤ cuH q1 uux L∞ ux H q ux L∞ uux H q L
≤ cu2H q1 ux L∞ uL∞ ,
in which we have used uux H q ≤ cu2 x H q ≤ cuL∞ uH q1 .
4.12
8
Abstract and Applied Analysis
For the fouth term in 4.9, using uu2x x uu2x x − ux u2x results in
Λq uΛq uux uxx dx
≤ 1 Λq ux Λq uu2x dx
2
R
R
1 q
q
2
u
Λ
uΛ
u
x x dx
2
R
4.13
K1 K2 .
For K1 , it follows from 4.12 that
4.14
K1 ≤ cu2H q1 ux L∞ uL∞ .
For K2 , applying Lemma 4.2 derives
K2 ≤ cuH q ux u2x Hq
≤ cuH q ux L∞ u2x Hq
ux H q u2x L∞
4.15
≤ cu2H q1 ux 2L∞ .
For the last term in 4.9, using Lemma 4.1 repeatedly results in
2
N−1
N
Λq ux Λq ux N dx
≤ cux q H ux q ≤ cuH q1 ux L∞ .
H
4.16
R
It follows from 4.10–4.16 that there exists a constant c such that
1 d
2 dt
R
.
Λq u2 Λq ux 2 dx ≤ cu2H q1 ux L∞ uL∞ ux 2L∞ ux N−1
L∞
4.17
Integrating both sides of the above inequality with respect to t results in inequality 4.6.
−1
To estimate the norm of ut , we apply the operator 1 − ∂2x to both sides of the first
equation of system 2.2 to obtain the equation
−1 4 1 ut 1 − ∂2x
− u3 ∂3x u3 − 2∂x uu2x uux uxx β∂x ux N .
x
3
3
4.18
Applying Λq ut Λq to both sides of 4.18 for q ∈ 0, s − 1 gives rise to
2
Λ ut dx q
R
Λ ut Λ
q
R
q−2
4 3 1 2 3
N
2
∂x − u ∂x u − 2uux βux uux uxx dτ.
3
3
4.19
Abstract and Applied Analysis
9
For the right-hand of 4.19, we have
Λ ut 1 −
q
R
∂2x
−1
≤ cut H q
R
Λ ∂x
q
4 3
2
− u − 2uux dx
3
2 1/2
q−1 4 2
1ξ
×
− u ξ−η u
η −2uu
x ξ−η u
x η dη
3
R
2
4.20
≤ cut H q uH 1 uH q1 uL∞ .
Since
−1
2
q 2
2
Λ ut 1 − ∂x Λ ∂x u ux dx − Λq ut Λq u2 ux dx
q
4.21
−1 Λ ut 1 − ∂2x Λq u2 ux dx,
q
using Lemma 4.2, u2 ux H q ≤ cu3 x H q ≤ cu2L∞ uH q1 , and uL∞ ≤ uH 1 , we have
Λq ut Λq u2 ux dx ≤ cut H q u2 ux Hq
≤ cut H q uL∞ uH 1 uH q1 ,
4.22
−1 Λq ut 1 − ∂2x Λq u2 ux dx
≤ cut q u ∞ u 1 u q1 .
H
L
H
H
Using the Cauchy-Schwartz inequality and Lemmas 4.1 and 4.2 yields
−1
N−1
Λq ut 1 − ∂2x Λq ∂x uN
x dx
≤ cut H q ux L∞ uH q1 ,
−1
Λq ut 1 − ∂2x Λq uux uxx dx
4.23
R
≤ cut H q uux uxx H q−2
≤ cut H q uu2x x q−2 ≤ cut H q u u2x
− ux u2x ≤ cut H q uu2x H
H q−1
ux u2x H q−2
x
≤ cut H q uu2x 4.24
H q−2
Hq
ux u2x Hq
≤ cut H q uH q1 uL∞ ux L∞ ux 2L∞ ,
in which we have used inequality 4.15.
Applying 4.20–4.24 into 4.19 yields the inequality
ut H q ≤ cuH q1 uL∞ uH 1 uL∞ ux L∞ ux 2L∞ ux N−1
L∞
for a constant c > 0. This completes the proof of Lemma 4.3.
4.25
10
Abstract and Applied Analysis
Defining
φx ⎧
⎨e1/x2 −1 ,
|x| < 1,
⎩0,
|x| ≥ 1,
4.26
and 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 R and s > 0.
It follows from Theorem 2.1 that for each ε the Cauchy problem
4 3
1
u
∂3x u3 − 2∂x uu2x uux uxx β∂x ux N
x
3
3
1
1
3 4
− u3 x ∂3x u3 − ∂x uu2x − u3x β∂x ux N ,
3
3
2
2
ut − utxx −
u0, x uε0 x,
4.27
x ∈ R,
has a unique solution uε t, x ∈ C∞ 0, T ; H ∞ .
Lemma 4.4. Under the assumptions of problem 4.27, the following estimates hold for any ε with
0 < ε < 1/4 and s > 0:
uε0x L∞ ≤ c1 u0x L∞ ,
uε0 H q ≤ c1 ,
if q ≤ s,
uε0 H q ≤ c1 εs−q/4 ,
uε0 − u0 H q ≤ c1 εs−q/4 ,
if q > s,
4.28
if q ≤ s,
uε0 − u0 H s o1,
where c1 is a constant independent of ε.
The proof of this Lemma can be found in Lai and Wu 7.
Lemma 4.5. If u0 x ∈ H s R with s ∈ 1, 3/2 such that u0x L∞ < ∞, and uε0 is defined as in
system 4.27. Then there exist two positive constants T and c, which are independent of ε, such that
the solution uε of problem 4.27 satisfies uεx L∞ ≤ c for any t ∈ 0, T .
Proof. Using notation u uε and differentiating both sides of the first equation of problem
4.27 with respect to x give rise to
1 2
3 2 1 3
N
N
2
3
−2
3
u
− βux .
utx uux u uxx u − βux − Λ u uux 2
2
2 x x
4.29
Abstract and Applied Analysis
11
Integrating by parts leads to
u2 uxx ux p1 dx −
R
1
p1
2p3
uux
dx,
integer p > 0,
4.30
R
from which we obtain
R
p−1
1 2
2p3
uux u2 uxx ux p1 dx uux dx.
2
2 p1 R
4.31
Multiplying the above equation by ux 2p1 and then integrating the resulting equation with
respect to x yield the equality
p−1
1 d
uux 2p3 dx
ux 2p2 dx 2p 2 dt R
2p 2 R
ux 2p1 u3 − βuN
x dx
R
4.32
−
3
1 3
ux − βuN
ux 2p1 Λ−2 u3 uu2x x dx.
x
2
2
R
Applying the Hölder’s inequality yields
1 d
ux 2p2 dx
2p 2 dt R
1/2p2 1/2p2 2p2 1/2p2
3
N 2p2
2p2
≤
dx
β
dx
dx
|G|
u ux R
R
|ux |2p2 dx
×
2p1/2p2
R
R
p−1 uux ∞
|ux |2p2 dx
L
2p 2 R
4.33
or
d
dt
ux 2p2 dx
1/2p2
R
1/2p2
1/2p2 1/2p2
3 2p2
N 2p2
2p2
≤
dx
β
dx
dx
|G|
u ux R
R
1/2p2
p−1 2p2
uux ∞
dx
,
|u
|
x
L
2p 2 R
R
4.34
12
Abstract and Applied Analysis
where
3
1 3
ux − βuN
G Λ−2 u3 uu2x x .
x
2
2
4.35
Since fLp → fL∞ as p → ∞ for any f ∈ L∞ L2 , integrating both sides of the inequality
4.34 with respect to t and taking the limit as p → ∞ result in the estimate
ux L∞ ≤ u0x L∞ t 2
c u3L∞ βux N
L∞ GL∞ uL∞ ux L∞ dτ.
4.36
0
Using the algebra property of H s0 R with s0 > 1/2 yields uε H 1/2 means that there exists
a sufficiently small δ > 0 such that uε H 1/2 uε H 1/2δ :
GL∞ ≤ cGH 1/2
−2 3 3 2 1 3 N Λ
uu
u
≤ c
−
βu
u
x
x
x x
2
2
H 1/2
≤ c u3H 1 Λ−2 uu2x 1/2 Λ−2 u3x x H 1/2
H
≤ c u3H 1 uu2x H −1
u3x H −1/2
uN
x ≤ c u3H 1 uux H −1 ux L∞ u3x H0
Λ−2 uN
x H 1/2
H0
ux N−1
L∞ uH 0
4.37
≤ u3H 1 u2H 1 ux L∞ uH 1 ux 2L∞ uH 1 ux N−1
,
L∞
in which Lemma 3.4 is used. From Lemma 4.3, we get
t
0
GL∞ dτ ≤ c
t
ec
τ
0
ux N−1
L∞ dξ
0
1 ux L∞ ux 2L∞ ux N−1
dτ.
L∞
4.38
Using uL∞ ≤ uH 1 , from 4.36 and 4.38, it has
ux L∞ ≤ u0x L∞ c
t
ec
t
0
ux N−1
L∞ dτ
0
N
1 ux L∞ ux 2L∞ ux N−1
L∞ ux L∞ dτ.
4.39
From Lemma 4.4, it follows from the contraction mapping principle that there is a
T > 0 such that the equation
WL∞ u0x L∞ c
t
0
ec
t
0
WN−1
L∞ dτ
N
1 WL∞ W2L∞ WN−1
L∞ WL∞ dτ
4.40
has a unique solution W ∈ C0, T . Using the theorem presented at page 51 in 8 yields that
there are constants T > 0 and c > 0 independent of ε such that ux L∞ ≤ Wt for arbitrary
t ∈ 0, T , which leads to the conclusion of Lemma 4.5.
Abstract and Applied Analysis
13
Using Lemmas 4.3 and 4.5, notation uε u, and Gronwall’s inequality results in the
inequalities
uε H q ≤ CT eCT ,
uεt H r ≤ CT eCT ,
4.41
where q ∈ 0, s, r ∈ 0, s − 1, and CT depends on T . It follows from Aubin’s compactness
theorem that there is a subsequence of {uε }, denoted by {uεn }, such that {uεn } and their temporal derivatives {uεn t } are weakly convergent to a function ut, x and its derivative ut in
L2 0, T , H s and L2 0, T , H s−1 , respectively. Moreover, for any real number R1 > 0, {uεn }
is convergent to the function u strongly in the space L2 0, T , H q −R1 , R1 for q ∈ 0, s and
{uεn t } converges to ut strongly in the space L2 0, T , H r −R1 , R1 for r ∈ 0, s − 1. Thus, we
can prove the existence of a weak solution to 1.2.
Proof of Theorem 2.2. From Lemma 4.5, we know that {uεn x } εn → 0 is bounded in the space
N
2
L∞ . Thus, the sequences {uεn } and {uN
εn x } are weakly convergent to u and ux in L 0, T ,
r
H −R, R for any r ∈ 0, s − 1, respectively. Therefore, u satisfies the equation
−
T 0
u gt − gxxt dx dt R
T 0
R
4 3 3 2
1
1
u uux gx − u3x gx − u3 gxxx − βux N gx dx dt
3
2
2
3
4.42
with u0, x u0 x and g ∈ C0∞ . Since X L1 0, T × R is a separable Banach space and
{uεn x } is a bounded sequence in the dual space X ∗ L∞ 0, T × R of X, there exists a
subsequence of {uεn x }, still denoted by {uεn x }, weakly star convergent to a function v in
L∞ 0, T × R. It derives from the {uεn x } weakly convergent to ux in L2 0, T × R that ux v
almost everywhere. Thus, we obtain ux ∈ L∞ 0, T × R.
Acknowledgment
This work is supported by the Applied and Basic Project of Sichuan Province 2012JY0020.
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