Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 216320, 13 pages
doi:10.1155/2012/216320
Research Article
The Asymptotic Solution of
the Initial Boundary Value Problem to a
Generalized Boussinesq Equation
Zheng Yin and Feng Zhang
Department of Applied Mathematics, Southwestern University of Finance and Economics,
Chengdu 610074, China
Correspondence should be addressed to Zheng Yin, yinzheng@swufe.edu.cn
Received 15 April 2012; Accepted 3 May 2012
Academic Editor: Shaoyong Lai
Copyright q 2012 Z. Yin and F. Zhang. 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 L2 space solution of an initial boundary problem for a generalized damped Boussinesq
equation is constructed. Certain assumptions on the coefficients of the equation are found to show
the existence and uniqueness of the global solution to the initial boundary problem. The explicit
expression for the large time asymptotic solution is obtained.
1. Introduction
The classical Boussinesq equation can be expressed by
utt −αuxxxx uxx β u2
,
xx
1.1
where ux, t is the elevation of the free surface of fluid, the subscripts denote partial
derivatives, and the constant coefficients α and β depend on the depth of fluid and the
characteristic speed of the long waves. The Boussinesq equation 1.1 was first derived by
Boussinesq 1, in 1872, to describe propagation of long waves with small amplitude on the
surface of shallow water. Since then, extensive research has been carried out to study the
property and solutions of the equation and its associated initial boundary value problems.
Clarkson 2 proposed a general approach for constructing exact solutions of 1.1. Hirota
3 deduced conservation laws and then examined the numerical solution of a Boussinesq
equation. Yajima 4 investigated the nonlinear evolution of a linearly stable solution for
1.1. The exponentially decaying solution of the spherical Boussinesq equation has been
2
Abstract and Applied Analysis
investigated by Nakamura 5. Galkin et al. 6 developed rational solutions of the onedimensional Boussinesq equation with both zero and nonzero boundary conditions at the
infinity in the space from the known solutions of the Kadomtzev-Petviashvili equation. The
structure of their solutions generalizes a family of rational solutions of the Korteweq-de
Vries equation to the case of two-wave processes. For many other methods to investigate
the Boussinesq system and other shallow water models, the reader is referred to 7–13 and
the references therein.
Various generalizations of the classical Boussinesq equation have been proposed and
studied from many aspects, in particular the well-posedness of the Cauchy problem for the
following equation,
utt −uxxxx uxx fu xx .
1.2
Bona and Sachs 14 showed that the special solitary wave solutions of 1.2 are nonlinearly
stable for a range of wave speeds. The local and global well-posedness of the problem has
been proved by transforming 1.2 into a system of nonlinear Schrödinger equations 15.
Varlamov 16 considered the initial-boundary condition for the following damped
Boussinesq equation:
,
utt − 2butxx −αuxxxx uxx β u2
xx
1.3
where the second term on the left-hand side is responsible for dissipation, β ∈ R1 , and α and
b are assumed to be positive constants and satisfy the assumption that α > b2 . An initialboundary value problem for 1.3 with small initial data is considered for the case of one
space dimension. The classical solution of the problem is constructed, and the long-time
asymptotics is obtained in explicit form. The asymptotics show the presence of both time
and space oscillations and the exponential decay of the solution in time due to dissipations.
Varlamov 17 has also used the eigenfunction expansion method to consider the long-time
asymptotics for a damped Boussinesq equation in a ball which is similar to 1.3.
The aim of this paper is to study the initial-boundary value problem for the following
generalized damped Boussinesq equation:
utt − auttxx − 2butxx 2b
ut −cuxxxx uxx − p2 u β u2
,
xx
a
1.4
where a, b and c are positive
constants and constants p ≥ 0 and β ∈ R1 . We assume the
√
2
cases a c > b and p ≥ 2b/a. The classical solutions of the initial-boundary value problem
for 1.4 will be constructed in the form of a Fourier series with coefficients in their own
term represented as series in terms of a small parameter involved in the initial conditions.
It is shown that the new solution of the initial-boundary value problem for 1.4 is wellposedness. In addition, the long time behavior of our solution also shows the presence of
damped oscillations decaying exponentially in time as t → ∞.
Abstract and Applied Analysis
3
2. Theorem of Existence, Uniqueness, and Asymptotics
In this paper, we consider the following initial-boundary value problem for the damped
Boussinesq equation:
utt − auttxx − 2butxx 2b
ut −cuxxxx uxx − p2 u β u2
,
xx
a
u0, t uπ, t 0,
uxx 0, t uxx π, t 0,
ux, 0 ε2 ϕx,
t > 0,
2.1
t > 0,
ut x, 0 ε2 ψx,
x ∈ 0, π,
where a, b, c, and p are positive constants, β ∈ R, and ε is a small parameter.
Definition 2.1. The function ux ∈ C2n 0, π, n ≥ 1, if u0 uπ u 0 u π · · · u2n−2 0 u2n−2 π 0 and u2n x ∈ L2 0, π.
Definition 2.2. The function ux, t defined on 0, π×0, ∞ is said to be the classical solution
of the problem defined by 2.1, if it has two continuous derivatives on 0, π × 0, ∞ and
satisfies system 2.1.
The main results obtained are summarized in the following theorem.
√
Theorem 2.3. If a > 0, a c > b2 , p ≥ 2b/a, ϕx ∈ C8 0, π, ψx ∈ C6 0, π, then there is a
ε0 > 0 such that, for 0 < ε ≤ ε0 , problem 2.1 has a unique classical solution of which is represented
in the form of
ux, t ∞
εN1 uN x, t,
2.2
N0
where the function uN x, t is as that defined by expressions 3.24 in the proof below. This series
and the series for the derivatives of ux, t involved in 2.1 converge absolutely and uniformly with
respect to x ∈ 0, π, t ≥ 0, ε ∈ 0, ε0 . In addition, the solution of system 2.1 has the following long
time behavior as t → ∞ :
ux, t e−b/at A cos σt B sin σt sin x O e−ηb/at ,
2.3
where σ ac a c − b2 1 ap2 − b2 /a2 p2 − b2 /a2 /1 a is a positive constant,
0 < η < 1, and the coefficients A and B are defined by 3.44.
3. Proof of Theorem 2.3
The proof of Theorem 2.3 includes three parts, namely, existence of a solution, uniqueness
of the solution, and the asymptotics of the solution. The main techniques used in this paper
are based on those presented in 16. However, it should be emphasized that the method for
proving uniqueness of the solution is different from that of 16.
4
Abstract and Applied Analysis
3.1. Existence of a Solution
We make an odd extension of ux, t in x to the interval −π, 0 and represent u in the form of
a complex Fourier series; namely,
∞
ux, t u
n teinx ,
u
n t n−∞
n/
0
π
1
2π
ux, te−inx dx,
−π
3.1
where u
−n t −
un t for n ≥ 1. For the above equalities, we shall use the fact that ux, t
belongs to the space L2 −π, π for each fixed t > 0, and we shall denote the corresponding
norm by
ut utL2 −π,π π
−π
1/2
.
3.2
ux, t sin nx dx.
3.3
2
|ux, t| dx
From 3.1, we obtain, through a simple calculation, that
ux, t 2i
∞
u
n t sin nx,
u
n t n1
1
iπ
π
0
Noting the initial functions, we can write on −π, π with
ϕ
−n ϕ
n ,
ψ
−n ψ
n ,
3.4
for n ≥ 1 that
ϕx ∞
ϕ
n einx ,
ψx n−∞
n/
0
∞
ψ
n einx .
n−∞
n/
0
3.5
Then on the interval 0, π, we obtain
ϕx 2i
∞
ϕ
n sin nx,
ψx 2i
n1
ϕ
n 1
iπ
∞
ψ
n sin nx,
n1
π
ϕx sin nx dx,
0
ψ
n 1
iπ
3.6
π
ψx sin nx dx.
0
Abstract and Applied Analysis
5
Integrating 3.6 by parts and using the smoothness assumption of the initial data, we get the
following inequalities:
ϕ
n ≤ C1 n−8 ,
ψ
n ≤ C1 n−6 ,
3.7
n ≥ 1,
where C1 is a positive constant. Substituting 3.1 and 3.5 into 2.1, we have the following
Cauchy problem for the function u
n t, n ∈ Z:
1 an
2
u
n t
b
2
u
n t cn4 n2 p2 u
n t −βn2 P un t,
2 bn a
u
n 0
u
n 0 ε ϕ
n ,
2
3.8
ε ψ
n ,
2
where
P un t ∞
u
n−g t
ug t,
u
−n t −
un t for n ≥ 1.
3.9
g−∞
g/
0,n
For n 1, we have
∞
∞
P u1 t 2 u
−g t
g t
u1g t −2 u
u1g t.
g1
3.10
g1
For n ≥ 2, it has
P un t n−1
∞
u
n−g t
g t
ug t − 2 u
ung t.
g1
3.11
g1
Letting Φn εϕ
n , Ψn εψ
n and solving the initial value problem 3.8, we get
b sinσn t
sinσn t
u
n t εe
Ψn
cosσn t ·
Φn a
σn
σn
t
βn2
b
exp
−
−
un τdτ,
−
τ
sin σn t − τP t
a
σn 1 an2 0
−b/at
3.12
where
σn acn6 a c − b2 n4 1 ap2 − 2b/a n2 p2 − b2 /a2 1 an2
n ≥ 1,
√
a c > b2 ,
p≥
,
2b
.
a
3.13
6
Abstract and Applied Analysis
Now, in order to solve the integral equation, we use the perturbation theory. Firstly, we
express u
n t, n ≥ 1, as a formal series in ε:
∞
u
n t εN1 ξ
n t.
N
3.14
N0
Then, by substituting 3.14 into 3.12 and equating the coefficients of the like powers of ε,
we get the following formulas, respectively, for N 0 and N ≥ 1 :
0
ξ
n t εe−b/at
N
ξ
n t −
βn2
σn 1 an2 t
0
b sinσn t
sinσn t
Ψn ,
cosσn t ·
Φn a
σn
σn
j b
exp − t − τ × sin σn t − τQN ξ
n τ dτ,
a
3.15
N ≥ 1, 3.16
where n ≥ 1 and
∞ N
N
n−1 j j−1
N−j
j−1
N−j
QN ξ
n τ εn
ξ
n−g τξ
g
ξ
ng τξ
g
τ − 2
τ,
g1 j1
3.17
g1 j1
in which ε1 0 and εn 1 for n ≥ 2.
Now we must prove that the formally constructed function 3.3, together with 3.12–
3.14, does represent a solution of problem 2.1. To do this, we shall show that the series
∞
ux, t einx
n−∞
n/
0
2i
∞
∞
εN1 ξ
n t
N
N0
sin nx
n1
∞
3.18
εN1 ξ
n τ
N
N0
converges absolutely and uniformly. For this purpose, we firstly establish the following time
estimates for n ≥ 1, t > 0, N ≥ 0:
N ξn t ≤ CN N 1−2 n−6 e−b/at .
3.19
Here and in the sequel we denote by C any positive constant independent of N, n, ε, and t,
but possibly depending on the coefficients of the equation and the initial functions.
We shall use the induction on the number N. For N 0, we have from 3.7 and 3.15
that
b 1
0 Φn |Ψn |
ξn t ≤ εe−b/at 1 aσn
σn
−6 −b/at
≤ εn e
.
3.20
Abstract and Applied Analysis
7
s
Assuming that 3.19 is valid for all ξ
n t with 0 ≤ s ≤ N − 1, we shall prove that 3.19 also
holds for s N. According to 16, for any integer n ≥ 1, g ≥ 1, and g /
n, we have
n − g −6 g −6 ≤ 26 n−6 g −6 n − g −6 ,
−2 −2
.
j −2 N 1 − j
≤ 22 N 1−2 j −2 N 1 − j
3.21
From 3.16 and noting that
bn2
b
≥
1 an2 a
3.22
for n / 0,
we have
∞ −6 N g −6 n − g ξn t ≤ CβN 1−2 n−6
g1
×
∞
−2
Cj−1 CN−j N 1 − j
j −2 |SN n, t|, |SN n, t|
j1
≤ Ce
−b/at
t
3.23
eb/a−2b/aτ dτ
0
≤ Ce
−b/at
eb/a−2b/at − 1 ×
b/a − 2b/a ≤ Ce−b/at .
Therefore, we know that inequality 3.19 holds.
N
For the derivation of 3.15, we recall 3.3 and 3.14 with ξ
n defined by 3.15 and
3.16 and interchange the order of summations in the series; namely,
ux, t 2i
∞
∞
∞
∞
N
u
n t sin nx 2i sin nx εN1 ξ
n t εN1 uN x, t,
n1
n1
N0
3.24
N0
where
uN x, t 2i
∞
N
ξ
n sin nx.
n1
3.25
8
Abstract and Applied Analysis
The interchange of the order of summations is allowable as the series is absolutely and
uniformly convergent for x ∈ 0, π, t ≥ 0 and ε ∈ 0, ε0 . Differentiating 3.15 and 3.16, we
get, for k 1, 2, that
0
∂tk ξ
n t
l
k
b sinσn t
sinσn t
l b
l
−b/at k−l
cosσn t Φn ck −1
e
∂k
Ψn ,
a
a σn
σn
l0
N
∂kt ξ
n t
βn2
−
1 an2 σn
t
gk n, t −
j
τQN ξ
n τ
3.26
dτ Rk n, t,
0
where
gk n, t k
ckl −1l
l0
l
b
k − lπ
−b/at k−l
,
e
σn sin σn t a
2
3.27
j
QN ξ
n τ is defined by 3.16, ckl l 0, 1, . . . , k are binomial coefficients, and Rk n, t k 1, 2 are obtained by differentiating the integral in 3.16 and are as follows:
R1 n, t 0,
j R2 n, t −βn2 QN ξ
n t .
3.28
Hence, using 3.19, we deduce that for, n ≥ 1, N ≥ 0, t > 0, and k ≥ 0, 1, 2,
k N ∂t ξn t ≤ CN N 1−2 n2k−6 e−b/at ,
k
n t ≤ Cn2k−6 e−b/at .
∂t u
3.29
Using these estimates and calculating the necessary derivatives of 3.29, we can prove
straightforwardly that 3.29 represents the classical solution of the initial-boundary value
problem defined by system 2.1.
3.2. Uniqueness of the Solution
For proving the uniqueness of the constructed solution, we shall assume that there exist two
classical solutions u1 x, t and u2 x, t to system 2.1 and then deduce that u1 x, t must
be equal to u2 x, t.
Abstract and Applied Analysis
9
Making an odd extension of the two solutions to the segment −π, 0, we notice that
both of them belong to the L2 −π, π space, and according to Definition 2.2, we have for each
fixed time t > 0 that
max u1 x, t ≤ Ct ,
max u2 x, t < Ct ,
x∈−π,π
3.30
x∈−π,π
where Ct is a constant depending on t. Let wx, t u1 x, t − u2 x, t, and make an even
extension of wx, t to . . . −3π, −2π, −2π, −π, π, 2π, 2π, 3π, . . . Then, we have from
2.1 that
wtt − awttxx − 2bwtxx 2b
wt −cwxxxx wxx − p2 w β wx, t u1 x, t u2 x, t
,
xx
a
wx, 0 wt x, t 0.
3.31
Taking the Fourier transform of w in the interval −∞, ∞; namely,
wξ,
t ∞
−∞
e−ixξ wx, tdx,
3.32
we have
b
t,
1 aξ2 w
ξ, t 2b ξ2 t −βξ2 fξ,
wξ,
t cξ4 ξ2 p2 wξ,
a
3.33
where
fx, t w u1 u2 x, t.
3.34
From 3.33, we have
wξ,
t −
βξ2
σξ 1 aξ2 t
b
τdτ,
exp − t − τ sin σξ t − τ fξ,
a
0
3.35
where
σξ acξ6 a c − bξ4 1 ap2 − 2b2 /a ξ2 p2 − b2 /a2 1 aξ2
.
3.36
10
Abstract and Applied Analysis
Thus, it has
t
b
t| ≤ C exp − t − τ fξ, τdτ
|wξ,
a
0
≤C
t
1/2
1/2 t 2
b
exp − t − τ dτ
fξ, τ dτ
a
0
0
3.37
t
1/2
2
≤C
.
fξ, τ dτ
0
It follows from 3.37 and the Parseval inequality that
∞
−∞
t|2 dξ ≤ C
|wξ,
∞ t 2
fξ, τ dτ dξ
−∞
0
t 2
t 2 dτ
≤ C fξ,
L
0
t 2
≤ C wx, τ u1 x, τ u2 x, τ 2 dτ
L
0
≤C
t
0
3.38
Cτ wx, τ2L2 dτ.
Using the Gronwall inequality, we obtain
wx, t 0
in L2 ,
3.39
which implies that u1 x, t ≡ u2 x, t and thus the solution is unique.
3.3. Long Time Asymptotics
To find the long-time behavior of the constructed solution, we firstly determine a subtle
asymptotic estimate of u
1 t which will contribute to the major term and then estimate the
∞
n t sin nx.
remaining terms n2 u
Since
u
1 t ∞
εN1 ξ
1
N0
N
t,
3.40
Abstract and Applied Analysis
11
we substitute it into the integral equations 3.15 and 3.16 to obtain
0
ξ
1 t e−b/at A0 cosσt B0 sinσt ,
N
ξ
1 t e−b/at
N
A
N
RA t
N
cosσt BN RB t sinσt ,
3.41
where
A0 ε
ϕ1 ,
b
ε
ϕ
1 ψ
B0 1 ,
σ 1a
√
ac 1 a c − b2
σ
,
1a
t
β
AN eb/aτ sinστSun τdτ,
σ1 a 0
t
β
N
−
eb/aτ cosστSun τdτ,
B
σ1 a 0
∞
β
N
eb/aτ sinστSun τdτ,
RA σ1 a 0
∞
β
N
eb/aτ cosστSun τdτ,
RB −
σ1 a 0
Sun t N
∞ j−1
N−j
ξ
tξ
g
t,
1g
3.42
N ≥ 1,
g1 j1
and the function ξ
j t, j 0, 1, . . . , N − 1 are defined by 3.15-3.16. Taking n ≥ 2, N ≥ 1
and using a method similar to that used in 16, it follows from 3.19 and 3.24 that there
exists a positive number 0 < η < 1 such that
N RA t ≤ ce−ηbt/a ,
N RB t ≤ ce−ηbt/a .
3.43
Hence, we have proved that, as t → ∞,
2i
u1 t e−b/at A cosσt B sinσt O e−1ηb/at ,
A 2i
∞
εN1 AN ,
N0
B 2i
∞
εN1 BN ,
3.44
N0
where 0 < η < 1, AN and BN are defined by 3.41, and the series above converge absolutely
and uniformly for ε ∈ 0, ε0 .
12
Abstract and Applied Analysis
Now, we can represent the solution by
ux, t 2i
u1 t Ru x, t,
Ru x, t 2i
∞
n2
sin nx
∞
N
ξ
n t.
3.45
N0
Using 3.24, we deduce that
∞
∞
−ηb
−ηb −2
N N1
−6
t
t ,
c ε
n ≤ c exp
N 1
|Ru x, t| ≤ exp
a
a
n0
N2
ux, t e−b/at A cos σt B sin σt sin x O e−ηb/at , 0 < η < 1.
3.46
3.47
It follows from 3.47 that the inequality 2.3 holds.
Acknowledgments
This work is supported by the Fundamental Research Funds for Central Universities SWUFE
2012.
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Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
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Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014