Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 840919, 23 pages
doi:10.1155/2012/840919
Research Article
The Cauchy Problem to a Shallow Water Wave
Equation with a Weakly Dissipative Term
Ying Wang and YunXi Guo
College of Science, Sichuan University of Science and Engineering, Zigong 643000, China
Correspondence should be addressed to Ying Wang, matyingw@126.com
Received 21 February 2012; Accepted 11 March 2012
Academic Editor: Shaoyong Lai
Copyright q 2012 Y. Wang and Y. Guo. 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.
A shallow water wave equation with a weakly dissipative term, which includes the weakly
dissipative Camassa-Holm and the weakly dissipative Degasperis-Procesi equations as special
cases, is investigated. The sufficient conditions about the existence of the global strong solution
are given. Provided that 1 − ∂2x u0 ∈ M R, u0 ∈ H 1 R, and u0 ∈ L1 R, the existence and
uniqueness of the global weak solution to the equation are shown to be true.
1. Introduction
The Camassa-Holm equation C-H equation
ut − utxx 3uux − 2ux uxx − uuxxx 0,
t > 0, x ∈ R,
1.1
as a model for wave motion on shallow water, has a bi-Hamiltonian structure and is
completely integrable. After the equation was derived by Camassa and Holm 1, a lot of
works was devoted to its investigation of dynamical properties. The local well posedness of
solution for initial data u0 ∈ H s R with s > 3/2 was given by several authors 2–4. Under
certain assumptions, 1.1 has not only global strong solutions and blow-up solutions 2, 5–
7 but also global weak solutions in H 1 R see 8–10. For other methods to handle the
problems related to the Camassa-Holm equation and functional spaces, the reader is referred
to 11–14 and the references therein.
To study the effect of the weakly dissipative term on the C-H equation, Guo 15 and
Wu and Yin 16 discussed the weakly dissipative C-H equation
ut − utxx 3uux − 2ux uxx − uuxxx λu − uxx 0,
t > 0, x ∈ R.
1.2
2
Abstract and Applied Analysis
The global existence of strong solutions and blow-up in finite time were presented in
16 provided that y0 1 − ∂2x u0 changes sign. The sufficient conditions on the infinite
propagation speed for 1.2 are offered in 15. It is found that the local well posedness
and the blow-up phenomena of 1.2 are similar to the C-H equation in a finite interval of
time. However, there are differences between 1.2 and the C-H equation in their long time
behaviors. For example, the global strong solutions of 1.2 decay to zero as time tends to
infinite under suitable assumptions, which implies that 1.2 has no traveling wave solutions
see 16.
Degasperis and Procesi 17 derived the equation D-P equation
ut − utxx 4uux − 3ux uxx − uuxxx 0,
t > 0, x ∈ R,
1.3
as a model for shallow water dynamics. Although the D-P equation 1.3 has a similar form to
the C-H equation 1.1, it should be addressed that they are truly different see 18. In fact,
many researchers have paid their attention to the study of solutions to 1.3. Constantin et
al. 19 developed an inverse scattering approach for smooth localized solutions to 1.3. Liu
and Yin 20 and Yin 21, 22 investigated the global existence of strong solutions and global
weak solutions to 1.3. Henry 23 showed that the smooth solutions to 1.3 have infinite
speed of propagation. Coclite and Karlsen 24 obtained global existence results for entropy
solutions in L1 R ∩ BV R and in L2 R ∩ L4 R.
The weakly dissipative D-P equation
ut − utxx 4uux − 3ux uxx − uuxxx λu − uxx 0,
t > 0, x ∈ R
1.4
is investigated by several authors 25–27 to find out the effect of the weakly dissipative term
on the D-P equation. The global existence, persistence properties, unique continuation and
the infinite propagation speed of the strong solutions to 1.4 are studied in 26. The blowup solution modeling wave breaking and the decay of solution were discussed in 27. The
1,∞
R ×R L∞
R ; H 1 R
existence and uniqueness of the global weak solution in space Wloc
loc
were proved see 25.
In this paper, we will consider the Cauchy problem for the weakly dissipative shallow
water wave equation
ut − utxx a buux − aux uxx − buuxxx λu − uxx 0,
u0, x u0 x,
x ∈ R,
t > 0, x ∈ R,
1.5
where a > 0, b > 0, and λ ≥ 0 are arbitrary constants, u is the fluid velocity in the x direction,
λu − uxx represents the weakly dissipative term. For λ 0, we notice that 1.5 is a special
case of the shallow water equation derived by Constantin and Lannes 28.
Since 1.5 is a generalization of the Camassa-Holm equation and the DegasperisProcesi equation, 1.5 loses some important conservation laws that C-H equation and DP equation possesses. It needs to be pointed out that Lai and Wu 12 studied global
existence and blow-up criteria for 1.5 with λ 0. To the best of our knowledge, the
dynamical behaviors related to 1.5 with λ 0, such as the global weak solution in space
1,∞
R × R L∞
R ; H 1 R, have not been yet investigated. The objective of this paper
Wloc
loc
is to investigate several dynamical properties of solutions to 1.5. More precisely, we firstly
Abstract and Applied Analysis
3
use the Kato theorem 29 to establish the local well-posedness for 1.5 with initial value
u0 ∈ H s with s > 3/2. Then, we present a precise blow-up scenario for 1.5. Provided that
u0 ∈ H s R L1 R and the potential y0 1−∂2x u0 does not change sign, the global existence
of the strong solution is shown to be true. Finally, under suitable assumptions, the existence
R ; H 1 R are proved. Our
and uniqueness of global weak solution in W 1,∞ R × R L∞
loc
main ideas to prove the existence and uniqueness of the global weak solution come from
those presented in Constantin and Molinet 8 and Yin 22.
2. Notations
The space of all infinitely differentiable functions φt, x with compact support in 0, ∞ × R
is denoted by C0∞ . Let 1 ≤ p < ∞, and let Lp Lp R be the space of all measurable functions
ht, x such that hPLP 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, let H s H s R denote the Sobolev
2
2 s 1/2
ξ space
< ∞, where ht,
−ixξwith the norm defined by hH s R 1 |ξ| |ht, ξ| dξ
e ht, xdx.
R
We denote by ∗ the convolution. Let · X denote the norm of Banach space X and
·, ·
the H 1 R, H −1 R duality bracket. Let MR be the space of the Radon measures on R
with bounded total variation and M R the subset of positive measures. Finally, we write
BV R for the space of functions with bounded variation, V f being the total variation of
f ∈ BV R.
−1
Note that if Gx : 1/2e−|x| , x ∈ R. Then, 1 − ∂2x f G ∗ f for all f ∈ L2 R and
G ∗ u − uxx u. Using this identity, we rewrite problem 1.5 in the form
ut buux ∂x G ∗
a 2 3b − a
u ux 2 λu 0,
2
2
u0, x u0 x,
t > 0, x ∈ R,
2.1
x ∈ R,
which is equivalent to
yt buyx ayux λy 0,
t > 0, x ∈ R,
2.2
y u − uxx ,
u0, x u0 x.
3. Preliminaries
Throughout this paper, let {ρn }n≥1 denote the mollifiers
−1
ρn x :
ρξdξ
R
nρnx,
x ∈ R, n ≥ 1,
3.1
4
Abstract and Applied Analysis
where ρ ∈ Cc∞ R is defined by
ρx :
⎧
⎨e1/x2 −1
for |x| < 1,
⎩0
for |x| ≥ 1.
3.2
Thus, we get
ρn xdx 1,
ρn ≥ 0, x ∈ R, n ≥ 1.
3.3
R
Next, we give some useful results.
Lemma 3.1 see 8. Let f : R → R be uniformly continuous and bounded. If μ ∈ MR, then
∞ 0 in L1 R.
3.4
ρn ∗ fμ − ρn ∗ f ρn ∗ μ n −→
Lemma 3.2 see 8. Let f : R → R be uniformly continuous and bounded. If g ∈ L∞ R, then
∞0
ρn ∗ fg − ρn ∗ f ρn ∗ g n −→
in L∞ R.
3.5
Lemma 3.3 see 30. Let T > 0. If f, g ∈ L2 0, T ; H 1 R and df/dt, dg/dt ∈
L2 0, T ; H −1 R, then f, g are a.e. equal to a function continuous from 0, T into L2 R and
ft, gt − fs, gs t t d fτ
d gτ
, gτ dτ , fτ dτ
dτ
dτ
s
s
3.6
for all s, t ∈ 0, T .
Lemma 3.4 see 8. Assume that ut, · ∈ W 1,1 R is uniformly bounded in W 1,1 R for all t ∈ R .
Then, for a.e. t ∈ R
d
ρn ∗ udx ρn ∗ ut sgn ρn ∗ u dx,
dt R
R
d
ρn ∗ ux dx ρn ∗ uxt sgn ρn ∗ ux dx.
dt R
R
3.7
4. Global Strong Solution
We firstly present the existence and uniqueness of the local solutions to the problem 2.1.
Theorem 4.1. Let u0 ∈ H s R, s > 3/2. Then, the problem 2.1 has a unique solution u, such that
u ut, x ∈ C0, T ; H s R ∩ C1 0, T ; H s−1 R ,
where T > 0 depends on u0 H s R .
4.1
Abstract and Applied Analysis
5
Proof. The proof of Theorem 4.1 can be finished by using Kato’s semigroup theory see 29
or 4. Here, we omit the detailed proof.
Theorem 4.2. Given u0 ∈ H s , s > 3/2, the solution u ·, u0 of problem 2.1 blows up in finite
time T < ∞ if and only if
−∞.
lim inf inf ux t, x
t→T
x∈R
4.2
Proof. Setting yt, x ut, x − uxx t, x, we get
2
2
y 2 u2 2u2x u2xx dx,
dx
−
u
u
xx
L
R
4.3
R
which yields
2
u2H 2 ≤ yL2 ≤ 2u2H 2 .
4.4
Using system 2.2, one has
d
2
y t, xdx 2 yyt dx
dt R
R
−2b uyyx dx − 2a ux y2 dx − 2λ y2 dx
R
R
−2a − b
ux y dx − 2λ
2
R
4.5
R
y2 dx.
R
Assume that there is a constant M > 0 such that
−ux t, x ≤ M
on 0, T × R.
4.6
From 4.5, we get
d
dt
y dx − 2λ
y t, xdx ≤ |2a − b|M
2
R
2
R
y2 dx.
4.7
R
Using Gronwall’ inequality, we deduce the uH 2 is bounded on 0, T . On the other hand,
−1
2
ut, x 1 − ∂x yt, x Gx − sysds.
4.8
R
Therefore, using 4.4 leads to
1 ux L∞ ≤ Gx x − sysds ≤ Gx L2 yL2 yL2 ≤ uH 2 .
2
R
4.9
It shows that if uH 2 is bounded, then ux L∞ is also bounded. This completes the proof.
6
Abstract and Applied Analysis
We consider the differential equation
qt bu t, q ,
t ∈ 0, T , x ∈ R,
q0, x x,
x ∈ R,
4.10
where u solves 1.5 and T > 0.
Lemma 4.3. Let u0 ∈ H s R s > 3; T is the maximal existence time of the corresponding solution
u to 1.5. Then, system 4.10 has a unique solution q ∈ C1 0, T × R; R. Moreover, the map qt, ·
is an increasing diffeomorphism of R with
qx t, x exp
t
bux s, qs, x ds
> 0,
∀t, x ∈ 0, T × R,
0
y t, qt, x qx2 t, x y0 x exp
t
−a − 2bux s, qs, x − λ ds .
4.11
0
Proof. From Theorem 4.1, we have u ∈ C1 0, T ; H s−1 R and H s−1 ∈ C1 R. We conclude
that both functions ut, x and ux t, x are bounded, Lipschitz in space, and C1 in time.
Applying the existence and uniqueness theorem of ordinary differential equations implies
that system 4.10 has a unique solution q ∈ C1 0, T × R, R.
Differentiating 4.10 with respect to x leads to
d
qx bux t, q qx ,
dt
t ∈ 0, T , b > 0,
qx 0, x 1,
4.12
x ∈ R,
which yields
qx exp
t
bux s, qs, x ds .
4.13
0
For every T < T , using the Sobolev embedding theorem gives rise to
sup
|ux s, x| < ∞.
s,x∈0,T ×R
4.14
It is inferred that there exists a constant K0 > 0 such that qx ≥ e−K0 t for t, x ∈ 0, T × R.
By computing directly, we derive
d y t, qt, x qx2 t, x −a − 2bux t, xyqx2 − λyqx2 ,
dt
4.15
Abstract and Applied Analysis
7
which results in
y t, qt, x
qx2 t, x
t
y0 x exp
−a − 2bux s, qs, x − λ ds .
4.16
0
The proof of Lemma 4.3 is completed.
Theorem 4.4. Let u0 ∈ L1 R ∩ H s R, s > 3/2, and 1 − ∂2x u0 ≥ 0 for all x ∈ R (or equivalently
1 − ∂2x u0 ≤ 0 for all x ∈ R). Then, problem 2.1 has a global strong solution
u u·, u0 ∈ C0, ∞; H s R ∩ C1 0, ∞; H s−1 R .
4.17
Moreover, if yt, · u − uxx , then one has for all t ∈ R ,
i yt, · ≥ 0, ut, · ≥ 0, and |ux t, ·| ≤ ut, · on R,
ii e−λt y0 L1 R yt, ·L1 R ut, ·L1 R e−λt u0 L1 R and ux t, ·L∞ R ≤
e−λt u0 L1 R ,
iii u2H 1 ≤ u0 2H 1 exp|a − 2b|/λ1 − e−λt u0 L1 − 2λt.
Proof. Let u0 ∈ H s , s > 3/2, and let T > 0 be the maximal existence time of the solution u
with initial date u0 cf. Theorem 4.1. If y0 ≥ 0, then Theorem 4.2 ensures that y ≥ 0 for all
t ∈ 0, T .
Note that u0 G ∗ y0 and y0 u0 − u0,xx ∈ L1 R. By Young’s inequality, we get
u0 L1 R G ∗ y0 t, ·L1 R ≤ GL1 R ≤ GL1 R y0 t, ·L1 R ≤ y0 L1 R .
4.18
Integrating the first equation of problem 2.1 by parts, we get
d
dt
u dx −b
R
uux dx −
R
a 2 3b − a
u ∂x G ∗
ux 2 dx λ u dx 0.
2
2
R
R
4.19
It follows that
u dx e
−λt
R
4.20
u0 dx.
R
Since y u − uxx , we have
y dx R
u dx −
R
e
−λt
uxx dx R
u0 dx e
R
−λt
u dx
R
u0 dx −
R
u0,xx dx
R
e
−λt
4.21
y0 dx.
R
8
Abstract and Applied Analysis
Given t ∈ 0, T , due to ut, x ∈ H s , s ≥ 3/2, from Theorem 4.1 and 4.21, we obtain
−ux t, x x
−∞
u dx ≤
x
−∞
∞
−∞
x
u − uxx dx ydx e
−λt
−∞
y dx
y0 dx e
−λt
4.22
R
u0 dx.
R
Note that u G ∗ y, y ≥ 0 on 0, T and the positivity of G. Thus, we can infer that u ≥ 0 on
0, T . From 4.22 we have
ux t, x ≥ −e−λt
u0 dx,
∀t, x ∈ 0, T × R.
4.23
R
From Theorem 4.2 and 4.23, we find T ∞. This implies that problem 2.1 has a unique
solution
u u·, u0 ∈ C0, ∞; H s R
C1 0, ∞; H s−1 R ,
s≥
3
.
2
4.24
Due to yt, x ≥ 0 and ut, x ≥ 0 for all t ≥ 0, it shows that
ut, x e−x
2
x
−∞
−x
e
ux t, x −
2
x
eξ yξdξ ∞
ex
2
x
x
e
e yξdξ 2
−∞
ξ
e−ξ yξdξ,
4.25
∞
−ξ
e yξdξ.
x
From the two identities above, we infer that ux 2 ≤ u2 on R for all t ≥ 0. This proves i.
Due to y ≥ 0, we obtain
ux t, x −
x
−∞
udx −
x
−∞
u − uxx dx −
x
−∞
ydx ≤ 0.
4.26
From u ≥ 0 and the inequality above, we get
ux t, x ≤
x
−∞
udx ≤
∞
−∞
udx e−λt
∞
−∞
u0 dx e−λt u0 L1 R .
On the other hand, from 4.23, we have that ux t, x ≥ −e−λt u0 L1 R . This proves ii.
4.27
Abstract and Applied Analysis
9
Multiplying the first equation of problem 2.1 by u and integrating by parts, we find
1 d
2 dt
u R
2
u2x
dx −a b
−λ
R
u ux dx a
uux uxx dx b
2
R
R
u2 uxxx dx
R
u2 u2x dx
a − 2b
−
2
R
u3x dx
−λ
4.28
R
u2 u2x dx,
which yields
R
u 2
u2x
dx −a − 2b
R
u3x dx
− 2λ
R
|ux |3 dx − 2λ
≤ |a − 2b|
R
≤ |a −
≤ |a −
R
2b|ux L∞ ux 2L2
2b|ux L∞ u2H 1
u2 u2x dx
− 2λ
u2 u2x dx
u R
− 2λ
2
u2x
4.29
dx
R
u2 u2x dx.
From Gronwall’s inequality, one has
u2H 1
≤
u0 2H 1
|a − 2b| −λt
1−e
exp
u0 L1 − 2λt .
λ
4.30
This proves iii and completes the proof of the theorem.
5. Global Weak Solution
Theorem 5.1. Let u0 ∈ H 1 R L1 R and y0 u0 − u0xx ∈ M R. Then equation 1.5 has
R ; H 1 R with initial data u0 u0 and such that
a unique solution u ∈ W 1,∞ R × R L∞
loc
u − uxx ∈ M , a.e. ∈ R is uniformly bounded on R.
Proof. We split the proof of Theorem 5.1 in two parts.
Let u0 ∈ H 1 R and y0 u0 −u0,xx ∈ M R. Note that u0 G∗y0 . Thus, for ϕ ∈ L∞ R,
we have
sup
ϕx G ∗ y0 xdx
ϕL∞ R ≤1 R
ϕx Gx − ξdy0 ξdx
u0 L1 R G ∗ y0 L1 R sup
ϕL∞ R ≤1
R
R
10
Abstract and Applied Analysis
sup
G ∗ ϕ ξdy0 ξ
ϕL∞ R ≤1 R
sup G ∗ ϕL∞ R y0 MR
ϕL∞ R ≤1
≤ sup GL1 R ϕL∞ R y0 MR y0 MR .
ϕL∞ R ≤1
5.1
Let us define un0 : ρn ∗ u0 ∈ H ∞ R for n ≥ 1. Obviously, we get
n
u 0
H 1 R
un0 −→ u0 in H 1 R for n −→ ∞,
ρn ∗ u0 H 1 R ρn ∗ u0 L2 ρn ∗ u0 L2 ≤ u0 H 1 R ,
5.2
un0 L1 R ρn ∗ u0 L1 R ≤ u0 L1 R .
Note that, for all n ≥ 1,
y0n : un0 − un0,xx ρn ∗ y0 ≥ 0.
5.3
Referring to the proof of 5.1, we have
n
y 0
L1 R
≤ y0 MR ,
n ≥ 1.
5.4
From the Theorem 4.4, we know that there exists a global strong solution
un un ·, un0 ∈ C0, ∞; H s R ∩ C1 0, ∞; H s−1 R ,
s≥
3
,
2
5.5
and un t, x − unxx t, x ≥ 0 for all t, x ∈ R × R.
Note that for all t, x ∈ R × R
un 2 x
−∞
2un unx dξ ≤
un 2 unx 2 dξ un 2H 1 .
5.6
R
From Theorem 4.4 and 5.2, we obtain
unx 2L∞ R ≤ un 2L∞ R ≤ un 2H 1 R
n 2
|a − 2b| −λt
n
1−e
u0 L1 − 2λt
≤ u0 H 1 exp
λ
|a − 2b| 2
−λt
≤ u0 H 1 exp
1−e
u0 L1 − 2λt .
λ
5.7
Abstract and Applied Analysis
11
From the Hölder inequality, Theorem 4.4, and 5.2, for all t ≥ 0 and n ≥ 1, we have
bun tunx tL2 R ≤ bun tL∞ R unx tL2 R ≤ bun 2H 1 R
2
|a − 2b| 1 − e−λt un0 L1 − 2λt
≤ bun0 H 1 exp
λ
|a − 2b| ≤ bu0 2H 1 exp
1 − e−λt u0 L1 − 2λt .
λ
5.8
Using Young’s inequality, we get
∂x G ∗ a un 2 3b − a unx 2 2
2
2
L R
a
3b a 2
2
∂x G ∗ un 2 ∂x G ∗ unx 2
L R
L R
2
2
a
3b a
≤ ∂x GL2 R un 2 1 ∂x GL2 R unx 2 1
L R
L R
2
2
≤
3b a
≤
∂x GL2 R un 2H 1 R
2
n 2
3b a
|a − 2b| −λt n 1−e
u0 L1 − 2λt
≤
∂x GL2 R u0 H 1 exp
2
λ
3b a
|a − 2b| 2
−λt
≤
1−e
u0 L1 − 2λt ,
∂x GL2 R u0 H 1 exp
2
λ
5.9
where ∂x GL2 R is bounded, and
λun L2 ≤ λun H 1
≤ λu0 H1
|a − 2b| −λt
1−e
exp
u0 L1 − λt .
2λ
5.10
Applying 5.8–5.10 and problem 2.1, we have
d n
u L2 R ≤
dt
3b a
|a − 2b| 1 − e−λt u0 L1 − 2λt
∂x GL2 R u0 2H 1 exp
2
λ
|a − 2b|
λu0 H 1 exp
1 − e−λt u0 L1 − λt .
2λ
b
5.11
For fixed T > 0, from 5.7 and 5.11, we deduce
T 0
R
2 un t, x2 unx t, x2 unt t, x dxdt ≤ M,
5.12
12
Abstract and Applied Analysis
where M is a positive constant depending only on Gx L2 R , u0 H 1 R , u0 L1 R , and T . It
follows that the sequence {un }n≥1 is uniformly bounded in the space H 1 0, T × R. Thus, we
can extract a subsequence such that
unk u,
weakly in H 1 0, T × R for nk −→ ∞,
unk −→ u,
a.e. on 0, T × R for nk −→ ∞,
5.13
5.14
for some u ∈ H 1 0, T × R. From Theorem 4.4 and 5.2, for fixed t ∈ 0, T , we have that the
sequence unxk t, · ∈ BV R satisfies
V unxk t, x unxxk t, ·L1 R ≤ unk t, ·L1 R ynk t, ·L1 R
≤ 2e−λt un0 k t, ·L1 R ≤ 2e−λt u0 t, ·L1 R ≤ 2e−λt y0 t, ·MR ,
5.15
nk
ux t, · ∞ ≤ e−λt unk 1 ≤ e−λt u0 1 ≤ e−λt y0 .
L R
0
L
MR
L R
Applying Helly’s theorem 31, we infer that there exists a subsequence, denoted again by
{unxk t, ·}, which converges at every point to some function νt, · of finite variation with
V νt, · ≤ 2e−λt y0 MR .
5.16
From 5.14, we get that for almost all t ∈ 0, T , unxk t, · → ux t, · in D R, it follows that
νt, · ux t, · for a.e. t ∈ 0, T . Therefore, we have
unxk t, · −→ ux t, ·
a.e. on 0, T × R for nk −→ ∞,
5.17
and, for a.e. t ∈ 0, T ,
V ux t, · uxx t, ·MR 2e−λt u0 L1 ≤ 2e−λt y0 MR .
By Theorem 4.4 and 5.7, we have
a n 2 3b − a n 2 u u
x 2
2
L2 R
≤
a
3b a n 2
n 2
u 2 ux 2
L R
L
2
2
≤
a n
3b a n
u L∞ un L2 R ux L∞ unx L2
2
2
5.18
Abstract and Applied Analysis
13
3b a n 2
a n 2
u H 1 R u H 1 R
2
2
3b
a
un 2H 1 R
2
3b
|a − 2b| 1 − e−λt xu0 L1 − 2λt .
≤ a
u0 2H 1 exp
2
λ
≤
5.19
Note that for fixed t ∈ 0, T , the sequence {a/2un 2 3b − a/2unx 2 }n≥1 is uniformly
2
bounded in L2 R. Therefore, it has a subsequence {a/2unk 2 3b − a/2unxk }nk ≥1 ,
which converges weakly in L2 R. From 5.14, we infer that the weak L2 R-limit is
{a/2u2 3b − a/2ux 2 }. It follows from Gx ∈ L2 R that
∂x G ∗
a nk 2 3b − a nk 2
ux
u 2
2
−→ ∂x G ∗
a
3b − a
u2 ux 2
2
2
for nk −→ ∞.
5.20
From 5.14, 5.17, and5.20, we have that u solves 2.1 in D 0, T × R.
For fixed T > 0, note that unt k is uniformly bounded in L2 R as t ∈ 0, T and
nk
u tH 1 R is uniformly bounded for all t ∈ 0, T and n ≥ 1, and we infer that the family
t → unk ∈ H 1 R is weakly equicontinous on 0,T. An application of the Arzela-Ascoli
theorem yields that {unk } has a subsequence, denoted again {unk }, which converges weakly
in H 1 R, uniformly in t ∈ 0, T . The limit function is u. T being arbitrary, we have that u is
locally and weakly continuous from 0, ∞ into H 1 R, that is, u ∈ Cw,loc R ; H 1 R.
Since, for a.e. t ∈ R , unk t, · ut, · weakly in H 1 R, from Theorem 4.4, we get
ut, ·L∞ R ≤ ut, ·H 1 R ≤ lim infunk t, ·H 1 R
≤ u0 H 1 exp
nk → ∞
|a − 2b| 1 − e−λt u0 L1 R − λt .
2λ
5.21
Inequality 5.21 shows that
∞
1
u ∈ L∞
loc R × R ∩ Lloc R ; H R .
5.22
From Theorem 4.4, 5.1 and 5.2, for t ∈ R , we obtain
unx t, ·L∞ ≤ e−λt un0 L1 R ≤ e−λt u0 L1 R ≤ e−λt y0 MR .
5.23
Combining with 5.14, we have
ux ∈ L∞ R × R.
Next, we will prove that
R
ut, ·dx e−λt
R
5.24
u0, ·dx by using a regularization approach.
14
Abstract and Applied Analysis
Since u satisfies 2.1 in distribution sense, convoluting 2.1 with ρn , we have that, for
a.e. t ∈ R ,
ρn ∗ ut ρn ∗ buux ρn ∗ ∂x p ∗
a 2 3b − a
u ux 2 λρn ∗ u 0.
2
2
5.25
Integrating the above equation with respect to x on R, we obtain
d
dt
ρn ∗ udx R
ρn ∗ buux dx
R
a 2 3b − a
u ρn ∗ ∂x p ∗
ux 2 dx λ ρn ∗ u dx 0.
2
2
R
R
5.26
Integration by parts gives rise to
d
dt
ρn ∗ udx −λ
R
ρn ∗ udx,
t ∈ R , n ≥ 1.
5.27
R
Utilizing Lemma 3.3, we obtain that
ρn ∗ ut, ·dx e
−λt
R
ρn ∗ u0 dx.
5.28
R
Since
lim ρn ∗ ut, · − ut, ·L1 R lim ρn ∗ u0 − u0 L1 R 0.
n→∞
n→∞
5.29
it follows that, for a.e. t ∈ R ,
ρn ∗ ut, ·dx lim e−λt
ut, ·dx lim
R
n→∞
n→∞
R
ρn ∗ u0 dx e−λt
R
u0 dx.
5.30
R
Finally, we prove that ut, · − uxx t, · ∈ M is uniformly bounded on R and ut, x ∈
W 1,∞ R × R.
Due to
L1 R ⊂ L∞ ∗ ⊂ C0 R∗ MR,
5.31
from 5.18, we get that, for a.e. t ∈ R ,
ut, · − uxx t, ·MR ≤ ut, ·L1 R uxx t, ·MR
≤ e−λt u0 L1 R 2e−λt y0 MR ≤ 3e−λt y0 MR .
5.32
Abstract and Applied Analysis
15
The above inequality implies that, for a.e. t ∈ R , ut, · − uxx t, · ∈ MR is uniformly
bounded on R. For fixed T ≥ 0, applying 5.13 and 5.14, we have
nk
u t, · − unxxk t, · −→ ut, · − uxx t, ·
in D R for n −→ ∞.
5.33
Since unk t, · − unxxk t, · ≥ 0 for all t, x ∈ R × R, we obtain that for a.e. t ∈ R , ut, · −
uxx t, · ∈ M R.
Note that ut, x G ∗ ut, x − uxx t, x. Then we get
|ut, x| |G ∗ ut, x − uxx t, x| ≤ GL∞ R ut, x − uxx t, xMR
≤ 3e−λt y0 MR .
5.34
Combining with 5.24, it implies that ut, x ∈ W 1,∞ R × R.
This completes the proof of the existence of Theorem 5.1.
Next, we present the uniqueness proof of the Theorem 5.1.
R ; H 1 R be two global weak solutions of problem 2.1
Let u, v ∈ W 1,∞ R ×R∩L∞
loc
with the same initial data u0 . Assume that ut, · − uxx t, · ∈ M R and vt, · − vxx t, · ∈
M R are uniformly bounded on R and set
!
N : sup ut, · − uxx t, ·MR vt, · − vxx t, ·MR .
5.35
t∈R
From assumption, we know that N < ∞. Then, for all t, x ∈ R × R,
|ut, x| |G ∗ ut, x − uxx t, x|
≤ GL∞ R ut, x − uxx t, xMR ≤
N
,
2
5.36
|ux t, x| |Gx ∗ ut, x − uxx t, x|
≤ Gx L∞ R ut, x − uxx t, xMR ≤
N
.
2
5.37
Similarly,
|vt, x| ≤
N
,
2
|vx t, x| ≤
N
,
2
t, x ∈ R × R.
5.38
Following the same procedure as in 5.1, we may also get that
ut, xL1 G ∗ ut, x − uxx t, xL1 R
≤ GL1 R ut, x − uxx t, xMR ≤ N,
ux t, xL1 R Gx ∗ ut, x − uxx t, xL1 R
≤ Gx L1 R ut, x − uxx t, xMR ≤ N
5.39
16
Abstract and Applied Analysis
and, for all t, x ∈ R × R,
|vt, x| ≤ N,
t, x ∈ R × R.
|vx t, x| ≤ N,
5.40
We define
wt, x : ut, x − vt, x,
t, x ∈ R × R.
5.41
Convoluting 2.1 for u and v with ρn , we get that for a.e. t ∈ R and all n ≥ 1,
a 2 3b − a
2
ρn ∗ ut ρn ∗ buux ρn ∗ ∂x G ∗
u ux λρn ∗ u 0,
2
2
a 2 3b − a
2
v ρn ∗ vt ρn ∗ bvvx ρn ∗ ∂x G ∗
vx λρn ∗ v 0.
2
2
5.42
5.43
Subtracting 5.43 from 5.42 and using Lemma 3.4, integration by parts shows that, for a.e.
t ∈ R and all n ≥ 1
d
dt
ρn ∗ wdx R
ρn ∗ wt sgn ρn ∗ w dx
R
ρn ∗ wux sgn ρn ∗ w dx
−b
R
ρn ∗ vwx sgn ρn ∗ w dx
−b
R
a
−
2
ρn ∗ ∂x G ∗ wu v sgn ρn ∗ w dx
5.44
R
3b − a ρn ∗ ∂x G ∗ wx ux vx sgn ρn ∗ w dx
2
R
−λ
ρn ∗ w sgn ρn ∗ w dx.
−
R
Using 5.36–5.38 and Young’s inequality to the first term on the right-hand side of 5.44
yields,
ρn ∗ wux sgn ρn ∗ w dx
R
≤
ρn ∗ wux dx
R
≤
R
ρn ∗ wρn ∗ ux dx ρn ∗ wux − ρn ∗ w ρn ∗ ux dx
R
Abstract and Applied Analysis
ρn ∗ w dx ρn ∗ wux − ρn ∗ w ρn ∗ ux dx
≤ ρn ∗ ux L∞
R
≤ ρn L1 ux L∞
N
2
≤
R
ρn ∗ wdx R
ρn ∗ wdx R
17
ρn ∗ wux − ρn ∗ w ρn ∗ ux dx
R
ρn ∗ wux − ρn ∗ w ρn ∗ ux dx.
R
5.45
Similarly, for the second term and the third term on the right-hand side of 5.44, we have
ρn ∗ wx v sgn ρn ∗ w dx
R
ρn ∗ wx v dx
≤
R
ρn ∗ wx ρn ∗ vdx ≤
R
≤
N
2
ρn ∗ wx dx R
ρn ∗ wx v − ρn ∗ wx ρn ∗ v dx
R
ρn ∗ wx v − ρn ∗ wx ρn ∗ v dx,
R
ρn ∗ ∂x G ∗ wu v sgn ρn ∗ w dx
R
ρn ∗ G ∗ wx u vdx
≤
R
R
ρn ∗ G ∗ wu v dx
x
ρn ∗ wx u vdx 1 ρn ∗ wux vx dx
2
R
R
N N ≤
ρn ∗ wx dx ρn ∗ wdx
2 R
2 R
ρn ∗ wx u v − ρn ∗ wx ρn ∗ u v dx
1
≤
2
R
R
ρn ∗ wu v − ρn ∗ w ρn ∗ u v dx.
x
x
5.46
For the last term on the right-hand side of 5.44, we have
ρn ∗ ∂x G ∗ wx u vx sgn ρn ∗ w dx
R
≤
ρn ∗ ∂x G ∗ |wx ||ux | |vx |dx
R
18
Abstract and Applied Analysis
ρn ∗ ∂x G ∗ |wx |dx
≤N
R
≤ N∂x GL1 R
ρn ∗ |wx |dx
R
ρn ∗ wx dx N
≤N
R
ρn ∗ |wx | − ρn ∗ wx dx .
R
5.47
From 5.45–5.47, for a.e. t ∈ R and all n ≥ 1, we find
d
dt
ρn ∗ wdx ≤
R
a 2b
ρn ∗ wdx
Nλ
4
R
a 2bN ρn ∗ wx dx Rn t,
5.48
R
where
Rn t −→ 0 as t −→ ∞,
|Rn t| ≤ K,
n ≥ 1, t ∈ R ,
5.49
where K is a positive constant depending on N and the H 1 R-norms of u0 and v0.
In the same way, convoluting 2.1 for u and v with ρn,x and using Lemma 3.4, we get
that, for a.e. t ∈ R and all n ≥ 1,
d
dt
ρn ∗ wx dx R
ρn ∗ wxt sgn ρn,x ∗ w dx
R
ρn ∗ wx ux vx sgn ρn,x ∗ w dx
−b
R
ρn ∗ vxx w sgn ρn,x ∗ w dx
−b
R
ρn ∗ uwxx sgn ρn,x ∗ w dx
−b
R
ρn ∗ ∂xx G ∗
−
R
a
3b − a 2
ux − vx2 sgn ρn,x ∗ w dx
u2 − v2 2
2
ρn ∗ wx sgn ρn,x ∗ w dx.
−λ
R
5.50
Abstract and Applied Analysis
19
Using the identity ∂2x G ∗ g G ∗ g − g for g ∈ L2 R and Young’s inequality, we estimate the
forth term of the right-hand side of 5.50
3b − a a 2
2
2
2
u −v ux − vx dx ρn ∗ ∂xx G ∗
2
2
R
3b − a a 2
2
2
2
≤ ρn ∗ G ∗
u −v ux − vx dx
2
2
R
3b − a a 2
2
2
2
ρn ∗
u −v ux − vx dx
2
2
R
ρn ∗ a wu v 3b − a wx ux vx dx
≤ GL1 R 1
2
2
R
≤ a ρn ∗ wu v dx 3b a ρn ∗ wx ux vx dx
R
ρn ∗ wdx 3b aN
≤ aN
R
5.51
R
ρn ∗ wx dx Rn .
R
Using 5.36–5.38 and Young’s inequality to the first term on the right-hand side of 5.50
gives rise to
−b
ρn ∗ wx ux vx sgn ρn,x ∗ w dx
R
ρn ∗ wx ux vx dx
≤b
R
ρn ∗ wx ρn ∗ ux vx dx
≤b
5.52
R
ρn ∗ wx ux vx − ρn ∗ wx ρn ∗ ux vx dx
b
R
ρn ∗ wx dx Rn .
≤ bN
R
To treat the second term of the right-hand side of 5.50, we note that
b
ρn ∗ vxx w sgn ρn,x ∗ w dx
R
ρn ∗ w ρn ∗ vxx dx
≤b
R
b
R
ρn ∗ vxx w − ρn ∗ w ρn ∗ vxx dx.
5.53
20
Abstract and Applied Analysis
Applying Lemma 3.1, the second expression of the right-hand side of 5.53 can be estimated
by a function Rn t belonging to 5.49. Making use of the Hölder inequality and 5.1, for a.e.
t ∈ R and all n ≥ 1, we have
ρn ∗ w ρn ∗ vxx dx ≤ ρn ∗ w
L∞ R
R
ρn ∗ vxx 1
L R
≤ ρn ∗ wW 1,1 R vxx MR .
5.54
It follows from 5.53 and 5.54 that
b
∗
v
w
sgn
ρ
∗
w
dx
ρ
n
xx
n,x
R
ρn ∗ wdx
≤ bN
5.55
R
bN
ρn ∗ wx Rn t.
R
Now, we deal with the third term on the right-hand side of 5.50
−b
ρn ∗ uwxx sgn ρn,x ∗ w dx
R
ρn ∗ u ρn ∗ wxx sgn ρn,x ∗ w dx
−b
R
ρn ∗ uwxx − ρn ∗ u ρn ∗ wxx sgn ρn,x ∗ w dx
−b
R
∂
ρn ∗ wx dx
≤ −b
ρn ∗ u
∂x
R
b ρn ∗ uwxx − ρn ∗ u ρn ∗ wxx dx
b
5.56
R
ρn ∗ ux ρn ∗ wx dx Rn .
R
Therefore, 5.56 implies that, for a.e. t ∈ R and all n ≥ 1,
−b
ρn ∗ uwxx sgn ρn,x ∗ w dx ≤ bN ρn ∗ wx dx Rn .
R
R
5.57
Abstract and Applied Analysis
21
From 5.51, 5.52, 5.55, and 5.57, for a.e. t ∈ R and all n ≥ 1, we deduce that
d
dt
ρn ∗ wx dx ≤ a bN
R
ρn ∗ wdx
R
ρn ∗ wx dx Rn .
6b aN λ
5.58
R
Combining with 5.48 and 5.58, we find
d
dt
ρn ∗ wdx
ρn ∗ w ρn ∗ wx dx ≤ 5a 6b N λ
4
R
R
2a 8bN λ ρn ∗ wx dx Rn
≤ 2a 8bN λ
5.59
R
ρn ∗ w ρn ∗ wx dx Rn .
R
It follows from Gronwall’ inequality that, for a.e. t ∈ R and all n ≥ 1,
ρn ∗ w ρn ∗ wx dx
R
" t
≤
#
ρn ∗ w ρn ∗ wx 0, xdx e2a8bNλt .
Rn sds 0
5.60
R
Fix t > 0, and let n → ∞ in 5.60. Since w u−v ∈ W 1,1 R and relation 5.49 holds, making
use of Lebesgue’s dominated convergence theorem yields
|w| |wx |0, xdx e2a8bNλt .
|w| |wx |dx ≤
R
5.61
R
Note that w0 wx 0 0; therefore, we obtain ut, x vt, x for a.e. t, x ∈ R × R. This
completes the proof of the theorem.
Acknowledgments
This work was supported by RFSUSE no. 2011KY12, RFSUSE no. 2011RC10 and the
project no. 12ZB080. The authors thanks the referee for valuable comments.
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