On Local Existence and Blow-Up of Solutions for Nonlinear Wave Equations of Higher-Order Kirchhoff Type with Strong Dissipation ()
1. Introduction
In this paper, we are concerned with local existence and blow-up of the solution for nonlinear wave equations of Higher-order Kirchhoff type with strong dissi- pation:
(1.1)
(1.2)
(1.3)
where
is a bounded domain in
with the smooth boundary
and
is the unit outward normal on
. Moreover,
is an integer constant, and
,
,
and
are some constants such that
,
,
,
and
. We call Equation (1.1) a non-degenerate equation when
and
, and a degenerate one when
and
. In the case of
and
, Equation (1.1) is usual semilinear wave equations.
It is known that Kirchhoff [1] first investigated the following nonlinear vib- ration of an elastic string for
:
(1.4)
where
is the lateral displacement at the space coordinate
and the time
;
: the mass density;
: the cross-section area;
: the length;
: the Young modulus;
: the initial axial tension;
: the resistance modulus; and
: the external force.
When
, the Equation (1.1) becomes a nonlinear wave equation:
(1.5)
(1.6)
(1.7)
It has been extensively studied and several results concerning existence and blowing-up have been established [2] [3] [4] .
When
, the Equation (1.1) becomes the following Kirchhoff equation with Lipschitz type continuous coefficient and strong damping:
(1.8)
(1.9)
(1.10)
where
is a bounded domain with a smooth boundary
. p > 2 and
is a positive local Lipschitz function. Here,![]()
. It has been studied and several results concerning existence and blowing-up have been established [5] .
When
, the Equation (1.1) becomes the following Kirchhoff equation:
(1.11)
(1.12)
(1.13)
where
is a bounded domain in
with the smooth boundary
and
is the unit outward normal on
. Moreover,
,
,
and
are some constants such that
,
,
,
and
. It has been studied and several results concerning existence and blowing-up have been established [6] .
When
, reference [7] has considered global existence and decay esti- mates for nonlinear Kirchhoff-type equation:
(1.14)
(1.15)
(1.16)
(1.17)
where
is a bounded domain of
with smooth boundary
such that
and
have positive measures, and
is the unit
outward normal on
, and
is the outward normal derivative on
.
The content of this paper is organized as follows. In Section 2, we give some lemmas. In Section 3, we prove the existence and uniqueness of the local solution by the Banach contraction mapping principle. In Section 4, we study the blow-up properties of solution for positive and negative initial energy and esti- mate for blow-up time
by lemma of [9] .
2. Preliminaries
In this section, we introduce material needed in the proof our main result. We use the standard Lebesgue space
and Sobolev space
with their usual scalar products and norms. Meanwhile we define
and introduce the following
abbreviations:
for any real number
.
Lemma 2.1 (Sobolev-Poincaré inequality [8] ) Let
be a number with
and
. Then there is a constant ![]()
depending on
and
such that
(2.1)
Lemma 2.2 [9] Suppose that
and
is a nonnegative
function such that
(2.2)
If
(2.3)
then we have
. Here,
is a constant and
the smallest positive root of the equation
![]()
Lemma 2.3 [9] If
is a non-increasing function on
such that
(2.4)
where
. Then there exists a finite time
such that
.
Moreover, for the case that
an upper bound of
is ![]()
If
, we have ![]()
If
, we have
or ![]()
3. Local Existence of Solution
Theorem 3.1 Suppose that
(
if
) and
for any given
, then there exists
such that the problem (1.1)-(1.3) has a unique local solution satisying
(3.1)
Proof. We proof the theorem by Banach contraction mapping principle. For
and
, we define the following two-parameter space of solutions:
(3.2)
where
. Then
is a complete metric space with the distance
(3.3)
We define the non-linear mapping
in the following way. For
is the unique solution of the following equation:
(3.4)
(3.5)
(3.6)
We shall show that there exist
and
such that
1)
maps
into itself;
2)
is a contraction mapping with respect to the metric
.
First, we shall check (i). Multiplying Equation (3.4) by
, and
integrating it over
, we have
(3.7)
where
.
To proceed the estimation,we observe that for
. By Lemma 2.1, we have
(3.8)
Because of
(
if
), then
(3.9)
Since
by the Young inequality, we see that
(3.10)
Combining these inequalities, we get
(3.11)
Therefore, by the Gronwall inequality, we obtain
(3.12)
where ![]()
and
(3.13)
So, for all
, we obtain
(3.14)
Therefore, in order that the map
verifies 1), it will be enough that the parameters
and
satisfy
(3.15)
Moreover, it follows from (3.14) that
and
. It implies
(3.16)
Next, we prove 2). Suppose that (3.15) holds. We take
, let
, and set
. Then
satisfies
(3.17)
(3.18)
(3.19)
(3.20)
Multiplying (3.17-3.18) by
and integrating it over
and using Green’s formula, we have
(3.21)
To proceed the estimation, by Lemma 2.1 observe that
(3.22)
(3.23)
where
.
(3.24)
Substituting (3.22)-(3.24) into (3.21), we obtain
(3.25)
According to the same method, Multiplying (3.17-3.18) by
and inte- grating it over
, we get
(3.26)
Taking (3.25)
(3.26) and by (3.10), it follows that
(3.27)
where ![]()
and
.
Applying the Gronwall inequality, we have
(3.28)
So, by (3.10) we have
(3.29)
where
. If
, we can see
is a contraction mapping. Finally, we choose suitable
is suffi- ciently large and
is sufficiently small, such that 1) and 2) hold. By applying Banach fixed point theorem, we obtain the local existence.
4. Blow-Up of Solution
In this section, we shall discuss the blow-up properties for the problem (1.1)- (1.3). For this purpose, we give the following definition and lemmas.
Now, we define the energy function of the solution
of (1.1)-(1.3) by
(4.1)
Then, we have
(4.2)
where ![]()
Definition 4.1 A solution
of (1.1)-(1.3) is called a blow-up solution, if there exists a finite time
such that
(4.3)
For the next lemma, we define
(4.4)
Lemma 4.1 Suppose that
(
if
) and
hold. Then we have the following results, which are
1)
, for t ≥ 0;
2) If
, we get
for
, where
;
3) If
and if
hold, then we
have
for
;
4) If
and
![]()
hold, then we get
for
.
Proof. Step 1: From (4.4), we obtain
(4.5)
and
(4.6)
From the above equation and the energy identity and
, we obtain
![]()
(4.7)
Therefore, we obtain 1).
Step 2: If
, then by (i), we have
(4.8)
Integrating (4.8) over
, we have that
(4.9)
Thus, we get
for
, where
.
So, 2) has been proved.
Step 3: If
, then for
we have
(4.10)
Integrating (4.10) over
, we have that
(4.11)
And because of
, then we get
.
Thus, 3) has been proved.
Step 4: For the case that
, we first note that
(4.12)
By using Hölder inequality, we have
(4.13)
So
(4.14)
Thus, we have
(4.15)
where ![]()
Set
(4.16)
Then
satisfies (2.2). By conditions
![]()
and Lemma 2.2, then
for
.
Lemma 4.2 Suppose that
(
if
) and
hold and that eigher one of the following conditions is satisfied:
1)
;
2)
and
;
3)
and
![]()
hold.
Then, there exists
, such that
for
.
Proof. By Lemma 4.1,
in case (i) and
in case 2) and 3).
Theorem 4.1 Suppose that
(
if
) and
hold and that eigher one of the following conditions is satisfied:
1)
;
2)
and
;
3)
and
![]()
hold.
Then the solution
blow up at finite
. And
can be estimated by (4.26)-(4.29), respectively, according to the sign of
.
Proof. Let
(4.17)
where
is some certain constant which will be chosen later. Then we get
(4.18)
and
(4.19)
where ![]()
By the Hölder inequality, we obtain
(4.20)
where
.
By 1) of Lemma 4.1, we get
(4.21)
Then, we obtain
(4.22)
Therefore, we get
(4.23)
Note that by Lemma 4.2,
Multiplying (4.23) by
and integrating it from
to
, we have
(4.24)
where
, and
.
When
and
, we obviously have
. When
,
we also have
by condition
.
Then by Lemma 2.3, there exists a finite time
such that ![]()
and the upper bounds of
are estimated respectively according to the sign of
. This will imply that
(4.25)
Next,
are estimated respectively according to the sign of
and Lemma 2.3.
In case 1), we have
(4.26)
Furthermore, if
, then we have
(4.27)
In case 2), we get
(4.28)
In case 3), we obtain
(4.29)
where
. Note that in case 1),
is given Lemma 4.1, and in
case 2) and case 3)
.
Remark 4.1 [10] The choice of
in (4.17) is possible under some conditions.
1) In the case
, we can choose
. In particular, we choose
, then we get
.
2) In the case
, we can choose
as in 1) if
or
if
.
3) For the case
. Under the condition
,
here
,
,
if
,
is chosen to satisfy
, where
,
Therefore, we have
.
5. Conclusion
In this paper, we prove that nonlinear wave equations of higher-order Kirchhoff Type with Strong Dissipation exist unique local solution on
. Then, we establish three blow-up results for certain solutions in the case 1):
, in the case 2):
and in the case 3):
. At last, we consider that the estimation of the upper bounds of the blow-up time
is given for deferent initial energy.
Acknowledgements
The authors express their sincere thanks to the anonymous reviewer for his/her careful reading of the paper, giving valuable comments and suggestions. These contributions greatly improved the paper.
This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11561076.