# Global Existence of Solution for Cauchy Problem of Two-Dimensional Boussinesq-Type Equation

Authors: Qingying Hu; Chenxia Zhang; Hongwei Zhang
Journal: ISRN Mathematical Analysis
 (2014)

---

## Abstract

In this paper, we consider the Cauchy problem of two-dimensional Boussinesq-type equationsu
t
t
-
Δ
u
-
Δ
u
t
t
+
Δ
2
u
=
Δ
f
u. Under the assumptions that f
u is a function with exponential growth at infinity and under some assumptions on the initial data, we prove the existence of global weak solution.


---

## Body

## 1. Introduction

In this paper, we study the following Cauchy problem of two-dimensional Boussinesq-type equations:(1)
u
t
t
-
Δ
u
-
Δ
u
t
t
+
Δ
2
u
=
Δ
f
(
u
)
,
x
∈
R
2
,
t
>
0
,
(2)
u
(
x
,
0
)
=
u
0
(
x
)
,
u
t
(
x
,
0
)
=
u
1
(
x
)
,
x
∈
R
2
,
where u
(
x
,
t
) denotes the unknown function, f
(
s
) is the given nonlinear function with exponential growth like e
s
2at the infinity, u
0
(
x
) and u
1
(
x
) are the given initial value functions, the subscript t indicates the partial derivative with respect to t, and Δ denotes the Laplace operator in R
2.This model arises in a number of mathematical models of physical processes, for example, in the modeling of surface waves in shallow waters or in considering the physical study of nonlinear wave propagation in waveguide [1–3]. In the one dimensional case, the longitudinal displacement u
(
x
,
t
) of the rod satisfies the following double dispersion equation (DDE) [1–3]:
(3)
u
t
t
-
u
x
x
=
1
4
(
6
u
2
+
a
u
t
t
-
b
u
x
x
)
x
x
,
and the general cubic DDE (CDDE),
(4)
u
t
t
-
u
x
x
=
1
4
(
c
u
3
+
6
u
2
+
a
u
t
t
-
b
u
x
x
+
d
u
t
)
x
x
,
where a
,
b
,
c, and d are positive constants. A great deal of efforts has been made to establish the sufficient condition for the existence or nonexistence of global solution to various nonlinear terms, such as f
(
s
)
=
±
a
|
s
|
p or f
(
s
)
=
a
|
s
|
p
s, p
>
1 or f
′
(
s
)
≥
C (see, e.g., [4–10]). In [7, 8] Chen et al. studied the initial boundary value problem and the Cauchy problem of the following generalized double dispersion equation which includes (4) as special cases:
(5)
u
t
t
-
u
x
x
-
a
u
x
x
t
t
+
b
u
x
x
x
x
+
d
u
x
x
t
=
f
(
u
)
x
x
.
For the case f
′
(
s
)
≥
C (bounded below) they proved the existence of global solutions. And they also show the nonexistence of the global solution under some other conditions to deal with the global well posedness of (4). Recently, in [9, 10], for the nonlinear term f
(
s
) satisfying more general conditions than both the convex function and f
(
s
)
=
|
s
|
p, the Cauchy problem and the initial boundary value problem for (5) with d
=
0, a
=
b
=
1 were studied, respectively. For both of above problems, the authors obtained the invariant sets and sharp conditions of global existence of solution by introducing a family of potential wells.But to the authors' best knowledge, there are very few works on the multidimensional cases for the Cauchy problems (1) and (2). Most recently, in [11–17], the authors considered the Cauchy problem of the multidimensional nonlinear evolution equation for constant k. Consider
(6)
u
t
t
-
Δ
u
-
Δ
u
t
t
+
Δ
2
u
-
k
Δ
u
t
=
Δ
f
(
u
)
,
x
∈
R
n
,
t
>
0
.
They gave the existence of local and global solution and the nonexistence of global solution. Note that, in [11–17], in order to obtain the global existence and the asymptotic of solution for problem (6), the authors requested that f
(
s
) possesses polynomial growth. In [14], the authors requested F
(
s
)
≥
0 or f
′
(
s
)
≥
C (bounded below), so the global existence of solution for Cauchy problem of (6) was solved. However, the results of [11–17] are not applicable for (1) with exponential growth like e
s
2 at the infinity. We point that the conditions of Lemma 2.2 and Corollary 2.3 in [14] are essentially power-type nonlinearity; that is, |
f
(
u
)
|
≤
a
|
u
|
p
+
b
|
u
|
q, where a and b are positive constants, although they requested F
(
s
)
≥
0 or f
′
(
s
)
≥
C (bounded below). In [11, 12], Polat et al. studied the existence, both locally and globally in time, asymptotic, and blow up of a solution for the problem (6) under the assumption f
′
(
s
)
≥
C and |
f
′
(
u
)
|
≤
a
|
u
|
p
+
b.In this paper, we consider the Cauchy problems (1) and (2) with nonlinear term like exponential growth at the infinity for two-dimensional case. The motivation of studying problems (1) and (2) with exponential growth nonlinear term comes from the following nonlinear damped wave equation:
(7)
u
t
t
-
Δ
u
+
h
(
u
t
)
=
g
(
u
)
,
in
Ω
×
(
0
,
+
∞
)
,
where Ω is a bounded domain of R
2. In the recent works of Ma and Soriano [18] and Alves and Cavalcanti [19], the authors assume that g admits an exponential growth and show global existence as well as blow up of solutions for problem (7) with initial-boundary conditions. The ingredients used in their proof [18, 19] are essentially the Trudinger-Moser inequality (see [20, 21]) and the well-known Mountain Pass Level by Ambrosetti and Rabinowitz [22].The main purpose of this paper is to establish the sufficient conditions for existence of global solution to the Cauchy problems (1) and (2) with exponential nonlinearities for spaces dimensions n
=
2, by considering similar arguments due to [18, 19]. The ingredients used in our proof are essentially the Trudinger-Moser inequality when Ω
=
R
2 (see [23, 24]) and Galerkin methods. As far as we are concerned, this is the first work in the literature that take into account the exponential growth of the function f for the Cauchy problems (1) and (2).This paper is organized as follows. Section2 is concerned with some notations and statement of assumptions. In Section 3, we prove the existence of global solutions to the Cauchy problems (1) and (2) by using Galerkin approximation scheme.


## 2. Preliminaries

In this paper, we denoteL
p
(
R
2
) by L
p
,
H
s
(
R
2
) by H
s, ∥
·
∥
p
=
∥
·
∥
L
p
(
R
2
), ∥
·
∥
=
∥
·
∥
L
2
(
R
2
), and (
u
,
v
)
=
∫
R
2
‍
u
(
x
)
v
(
x
)
d
x. We consider problems (1) and (2) with initial data u
0
∈
H
1, u
1
∈
L
2, and (
-
Δ
)
1
/
2
u
1
∈
L
2. Here, (
-
Δ
)
s
u
=
F
-
1
(
|
ξ
|
2
s
F
(
u
)
), s
>
0 and F
(
u
), F
-
1
(
u
) are the Fourier transformation and the inverse Fourier transformation, respectively. We also define the space
(8)
H
=
{
u
∈
H
1
∣
(
-
Δ
)
1
/
2
u
∈
L
2
}
,
with the norm
(9)
∥
u
∥
H
2
=
∥
u
∥
H
1
2
+
∥
(
-
Δ
)
1
/
2
u
∥
2
,
and the space
(10)
L
=
{
u
∈
L
2
∣
(
-
Δ
)
1
/
2
u
∈
L
2
}
,
with the norm
(11)
∥
u
∥
H
2
=
∥
u
∥
2
+
∥
(
-
Δ
)
1
/
2
u
∥
2
.
Since H
1 is dense in L
2, we have that H is dense in L.We callu
(
x
,
t
) a weak solution of problems (1) and (2) on R
2
×
[
0
,
T
), if u
∈
L
∞
(
0
,
T
;
H
1
), u
t
∈
L
∞
(
0
,
T
;
L
), satisfying
(12)
(
(
-
Δ
)
1
/
2
u
t
,
(
-
Δ
)
1
/
2
v
)
+
(
u
t
,
v
)
+
∫
0
t
‍
(
(
u
,
v
)
+
(
∇
u
,
∇
v
)
+
(
f
(
u
)
,
v
)
)
d
s
=
(
(
-
Δ
)
1
/
2
u
1
,
(
-
Δ
)
1
/
2
v
)
+
(
u
1
,
v
)
,
∀
v
∈
H
,
∀
t
∈
[
0
,
T
)
,
u
(
x
,
0
)
=
u
0
(
x
) in H
1, and
(13)
u
t
(
x
,
0
)
=
u
1
(
x
)
in
L
.Throughout this paper we will make repeated use of the Trudinger-Moser inequality which can be found in [23, 24].Proposition 1 (see [23, 24] (Trudinger-Moser inequality)).
Ifα
>
0 and u
∈
H
1
(
R
2
), then
(14)
∫
R
2
‍
(
e
α
|
u
|
2
-
1
)
d
x
∈
L
1
(
R
2
)
.
Moreover, if α
<
4
π, then there exists a constant l
>
0, such that
(15)
sup
⁡
∥
u
∥
H
1
≤
1
∫
R
2
‍
(
e
α
|
u
|
2
-
1
)
d
x
≤
l
<
∞
.
The nonlinear functionf is required to satisfy the following conditions:(H)
f
:
R
→
R
is a continuous function; for each
β
>
0
, there exists a positive constant
C
β
such that
(16)
|
f
(
t
)
|
≤
C
β
(
e
β
t
2
-
1
)
,
lim
⁡
t
→
0
f
(
t
)
t
=
0
,
and, for each
ε
>
0
,
β
>
0
, there exists a positive constant
a
, which depends on
ε
,
β
, verifying
(17)
|
f
(
t
)
|
≤
ε
|
t
|
+
a
(
ε
,
β
)
|
t
|
p
-
1
(
e
β
t
2
-
1
)
,
(18)
|
F
(
t
)
|
≤
ε
2
|
t
|
2
+
a
(
ε
,
β
)
|
t
|
p
(
e
β
t
2
-
1
)
,
(19)
f
(
t
)
t
≥
θ
F
(
s
)
>
0
,
t
∈
R
∖
{
0
}
,for some θ
>
2, where F
(
t
)
=
∫
0
t
‍
f
(
s
)
d
s.
As in [15], we introduce two functionals as follows:
(20)
E
(
t
)
=
E
(
u
)
=
1
2
(
∥
(
-
Δ
)
1
/
2
u
t
∥
2
+
∥
u
t
∥
2
+
∥
u
∥
H
1
2
)
+
∫
R
2
‍
F
(
u
)
d
x
,
E
(
0
)
=
1
2
(
∥
(
-
Δ
)
1
/
2
u
1
∥
2
+
∥
u
1
∥
2
+
∥
u
0
∥
H
1
2
)
+
∫
R
2
‍
F
(
u
0
)
d
x
.According to the Trudinger-Moser inequality (Proposition1), if f satisfies (H), then E
(
u
) and E
(
0
) are well defined on H.In order to prove our main results we also need the following inequality, for alls
∈
R and p
≥
1,
(21)
(
e
s
2
-
1
)
p
≤
e
p
s
2
-
1
.


## 3. Existence of Global Solution

In this section, we study the existence of global solution for problem (1)-(2). To this end, we begin this section by the following lemma.Lemma 2.
Suppose thatf
(
s
) satisfies the assumption (H) and assume that {
u
m
} is a bounded sequence in L
∞
(
0
,
T
;
H
1
). Then, if for some u,
(22)
u
m
⟶
u
w
e
a
k
l
y
i
n
L
2
(
0
,
T
;
L
2
(
R
2
)
)
;
there exists a subsequence {
u
k
} of {
u
m
}, which satisfies the following convergence:
(23)
f
(
u
k
)
⟶
f
(
u
)
w
e
a
k
l
y
i
n
L
2
(
0
,
T
;
L
2
(
R
2
)
)
,
(24)
F
(
u
k
)
⟶
F
(
u
)
w
e
a
k
l
y
i
n
L
1
(
0
,
T
;
L
1
(
R
2
)
)
.Proof.
Since{
u
m
} is bounded, let M
>
0 such that |
|
u
m
|
|
H
1
≤
M for all m
∈
N and t
∈
(
0
,
T
). Then selecting β
>
0 such that 4
β
M
2
<
π, we have from (17) and (21) that
(25)
∫
R
2
‍
|
f
(
u
m
)
|
2
d
x
≤
∫
R
2
‍
C
1
[
ε
2
|
u
m
|
2
+
a
2
|
u
m
|
2
(
p
-
1
)
(
e
2
β
u
m
2
-
1
)
]
d
x
≤
C
1
[
ε
2
∥
u
m
∥
2
+
a
2
∥
u
m
∥
4
(
p
-
1
)
2
(
p
-
1
)
(
∫
R
2
‍
(
e
4
β
u
m
2
-
1
)
d
x
)
1
/
2
]
.
Using Trudinger-Moser inequality (15), we obtain
(26)
sup
⁡
|
|
u
m
|
|
H
1
≤
M
∫
R
2
‍
(
e
4
β
u
m
2
-
1
)
d
x
=
sup
⁡
∥
u
m
∥
H
1
≤
M
∫
R
2
‍
(
e
4
β
∥
u
m
∥
H
1
2
(
u
m
/
∥
u
m
∥
H
1
)
2
-
1
)
d
x
≤
sup
⁡
∥
u
m
∥
H
1
≤
M
∫
R
2
‍
(
e
4
β
M
2
(
u
m
/
∥
u
m
∥
H
1
)
2
-
1
)
d
x
≤
l
.
Then, by (25), (26), and embedding theorem, we get
(27)
∫
R
2
‍
|
f
(
u
m
)
|
2
d
x
≤
C
(
ε
,
β
,
M
,
C
*
)
,
for all m. This shows that {
f
(
u
m
)
} is a bounded sequence in L
2
(
0
,
T
;
L
2
(
R
2
)
). Besides, since H
1
↪
L
2 is compact, there exists a subsequence {
u
k
} of {
u
m
}, still denoted by {
u
m
}, such that u
m
→
u a.e. in (
0
,
T
)
×
R
2; it follows from the continuity of f that f
(
u
m
)
→
f
(
u
) a.e. in (
0
,
T
)
×
R
2. Then (23) follows from a well-known result by Lions ([25], Ch. 1, Lemma 1.3).
As the above arguing, using (18), the Trudinger-Moser inequalities (15) and (21), we obtain
(28)
∫
R
2
‍
|
F
(
u
m
)
|
d
x
≤
C
(
ε
,
β
,
M
,
C
*
)
,
for all m. Then, (24) holds.
Now, we consider the existence of global solution for problems (1) and (2).
Letu
0
∈
H
,
u
1
∈
L, and {
w
j
}
j
=
1
∞ be basis function systems in H. Construct the approximate solutions of problems (1) and (2). Consider
(29)
u
m
(
x
,
t
)
=
∑
j
=
1
m
‍
g
j
m
(
t
)
w
j
(
x
)
,
m
=
1,2
,
…
,
satisfying
(30)
(
(
-
Δ
)
1
/
2
u
m
t
t
,
(
-
Δ
)
1
/
2
w
s
)
+
(
u
m
t
t
,
w
s
)
+
(
u
m
,
w
s
)
+
(
∇
u
m
,
∇
w
s
)
+
(
f
(
u
)
,
w
s
)
=
0
,
s
=
1,2
,
…
,
m
,
(31)
u
m
(
x
,
0
)
=
∑
j
=
1
m
‍
a
j
m
(
t
)
w
j
(
x
)
⟶
u
0
(
x
)
in
H
,
(32)
u
m
t
(
x
,
0
)
=
∑
j
=
1
m
‍
b
j
m
(
t
)
w
j
(
x
)
⟶
u
1
(
x
)
in
L
.
By multiplying (30) by g
s
m
′
(
t
) and summing for s, we get
(33)
d
d
t
E
m
(
t
)
=
0
,
(34)
E
m
(
t
)
=
E
m
(
0
)
,
where
(35)
E
m
(
t
)
=
E
(
u
m
)
=
1
2
(
∥
(
-
Δ
)
1
/
2
u
m
t
∥
2
+
∥
u
m
t
∥
2
+
∥
u
m
∥
H
1
2
)
+
∫
R
2
‍
F
(
u
m
)
d
x
.Lemma 3.
Suppose thatf
(
s
) satisfies the assumption (H), u
0
∈
H
,
u
1
∈
L, then F
(
u
0
)
∈
L
1 and E
m
(
0
)
→
E
(
0
) as m
→
∞. Furthermore, there exists positive constant E
0 which is independent of m such that
(36)
E
m
(
0
)
≤
E
0
.Proof.
Since (18), (14), and u
0
∈
H, we have
(37)
∫
R
2
‍
|
F
(
u
0
)
|
d
x
≤
ε
2
∫
R
2
‍
|
u
0
|
2
d
x
+
a
(
ε
,
β
)
×
(
∫
R
2
‍
|
u
0
|
2
p
d
x
)
(
∫
R
2
‍
(
e
2
β
u
0
2
-
1
)
d
x
)
;
then we get F
(
u
0
)
∈
L
1. It follows from (31) and (24) in Lemma 2 that
(38)
∫
R
2
‍
F
(
u
m
(
0
)
)
d
x
⟶
∫
R
2
‍
F
(
u
0
)
d
x
.
From (31) and (32), we get that, as m
→
∞,
(39)
∥
(
-
Δ
)
1
/
2
u
m
t
(
0
)
∥
2
+
∥
u
m
t
(
0
)
∥
2
+
∥
u
m
(
0
)
∥
H
1
2
⟶
∥
(
-
Δ
)
1
/
2
u
1
∥
2
+
∥
u
1
∥
2
+
∥
u
0
∥
H
1
2
.
By combining (38) with (39), we have E
m
(
0
)
→
E
(
0
) as m
→
∞ and then (36) is obvious.Theorem 4.
Letf
(
s
) satisfy the assumption (H), u
0
∈
H
,
u
1
∈
L. Then, for any T
>
0, the problems (1) and (2) admit a global weak solution u
(
t
)
∈
L
∞
(
0
,
∞
;
H
1
) with u
t
(
t
)
∈
L
∞
(
0
,
∞
;
L
).Proof.
For problems (1) and (2), construct the approximate solutions (30) and (31). From (34) and (36), we get
(40)
E
m
(
t
)
=
1
2
(
∥
(
-
Δ
)
1
/
2
u
m
t
∥
2
+
∥
u
m
t
∥
2
+
∥
u
m
∥
H
1
2
)
+
∫
R
2
‍
F
(
u
m
)
d
x
=
E
m
(
0
)
≤
E
0
.
From (40) and (19), it follows that {
u
m
} in L
∞
(
0
,
T
;
H
1
) and {
u
m
t
} in L
∞
(
0
,
T
;
L
) are bounded, respectively. Hence, by Lemma 2, there exists a u and a subsequence of {
u
m
}, still denoted by {
u
m
}, such that u
m
→
u in L
∞
(
0
,
T
;
H
1
) weakly star and a.e. in Q
=
R
2
×
[
0
,
∞
);
u
m
t
→
u
t in L
∞
(
0
,
T
;
L
) weakly star;
f
(
u
m
)
→
f
(
u
) in L
∞
(
0
,
T
;
L
2
) weakly star.By integrating (40) with respect to t from 0 to t, we get
(41)
(
(
-
Δ
)
1
/
2
u
m
t
,
(
-
Δ
)
1
/
2
w
j
)
+
(
u
m
t
,
w
j
)
+
∫
0
t
‍
(
(
u
m
,
w
j
)
+
(
∇
u
m
,
∇
w
j
)
+
(
f
(
u
m
)
,
w
j
)
)
d
s
=
(
(
-
Δ
)
1
/
2
u
m
t
(
0
)
,
(
-
Δ
)
1
/
2
w
j
)
+
(
u
m
t
(
0
)
,
w
j
)
.
Let m
→
∞ in (41), we obtain
(42)
(
(
-
Δ
)
1
/
2
u
t
,
(
-
Δ
)
1
/
2
w
j
)
+
(
u
t
,
w
j
)
+
∫
0
t
‍
(
(
u
,
w
j
)
+
(
∇
u
,
∇
w
j
)
+
(
f
(
u
)
,
w
j
)
)
d
s
=
(
(
-
Δ
)
1
/
2
u
1
,
(
-
Δ
)
1
/
2
w
j
)
+
(
u
1
,
w
j
)
,
∀
j
,
(
(
-
Δ
)
1
/
2
u
t
,
(
-
Δ
)
1
/
2
v
)
+
(
u
t
,
v
)
+
∫
0
t
‍
(
(
u
,
v
)
+
(
∇
u
,
∇
w
j
)
+
(
f
(
u
)
,
v
)
)
d
s
=
(
(
-
Δ
)
1
/
2
u
1
,
(
-
Δ
)
1
/
2
v
)
+
(
u
1
,
v
)
,
∀
v
∈
H
,
∀
t
∈
[
0
,
T
)
.
On the other hand, from (31) and (32) we have u
(
x
,
0
)
=
u
0
(
x
) in H
1 and u
t
(
x
,
0
)
=
u
1
(
x
) in L. Therefore, the above u
(
x
) is a global weak solution of problems (1) and (2).


## 4. Example and Generalization

In this section we give some examples of the conditions in Section2 and some generalizations for the results obtained in Section 3.A typical example off
(
t
) satisfying (H) is f
(
t
)
=
t
(
e
k
|
t
|
α
-
1
), for some k
≥
0, 1
<
α
<
2. Obviously, lim
⁡
t
→
0
(
f
(
t
)
/
t
)
=
0, and direct computations show that, for any β
>
0, lim
⁡
t
→
0
(
f
(
t
)
/
(
e
β
t
2
-
1
)
)
=
0 and lim
⁡
t
→
∞
(
f
(
t
)
/
(
e
β
t
2
-
1
)
)
=
0, since 1
<
α
<
2, then (16) and (17) hold. From (17), we get (18). For 2
<
θ
≤
3, we have, for t
≠
0,
(43)
0
<
θ
F
(
t
)
=
θ
∫
0
t
‍
s
(
e
k
|
s
|
α
-
1
)
d
s
=
θ
∫
0
t
‍
(
∑
j
=
0
∞
‍
k
j
|
s
|
α
j
+
1
j
!
-
s
)
d
s
=
θ
(
∑
j
=
0
∞
‍
k
j
|
t
|
α
j
+
2
j
!
(
α
j
+
2
)
-
1
2
t
2
)
=
θ
t
2
(
∑
j
=
0
∞
‍
k
j
|
t
|
α
j
j
!
(
α
j
+
2
)
-
1
2
)
=
θ
t
2
∑
j
=
1
∞
‍
k
j
|
t
|
α
j
j
!
(
α
j
+
2
)
≤
θ
α
+
2
t
2
∑
j
=
1
∞
‍
k
j
|
t
|
α
j
j
!
≤
t
f
(
t
)
.
Hence, (19) holds.Another example off
(
t
) satisfying (H) is f
(
t
)
=
|
t
|
p
-
2
t
(
e
k
|
t
|
α
-
1
), for some k
≥
0
,
p
≥
2
,
1
<
α
<
2. Obviously (16), (17), and (18) are satisfied.We also can takef
(
x
,
t
)
=
ϕ
(
x
)
t
(
e
k
|
t
|
α
-
1
), for some k
≥
0
,
1
<
α
<
2 and ϕ
(
x
) is a continuous function with 0
<
c
1
≤
ϕ
≤
c
2 for constants c
1 and c
2.Consider the following two-dimensional double dispersion equation with damped:(44)
u
t
t
-
Δ
u
-
Δ
u
t
t
+
Δ
2
u
-
Δ
u
t
=
Δ
f
(
u
)
,
x
∈
R
2
,
t
>
0
,
then (34) can be written as
(45)
E
m
(
t
)
+
∫
0
t
‍
∥
u
m
s
∥
2
d
s
=
E
m
(
0
)
;
we can get the following result.Theorem 5.
Letf
(
s
) satisfy the assumption (H), u
0
∈
H
,
u
1
∈
L. Then, for any T
>
0, the problems (44) and (2) admit a global weak solution u
(
t
)
∈
L
∞
(
0
,
∞
;
H
1
) with u
t
(
t
)
∈
L
∞
(
0
,
∞
;
L
).We can also consider the problems (44) and (2) in R
n for n
>
2 and get the similar results.



---

*Source: 890503-2014-03-27.xml*