International Journal of Modern Nonlinear Theory and Application
Vol.06 No.01(2017), Article ID:74071,15 pages
10.4236/ijmnta.2017.61002
On Local Existence and Blow-Up of Solutions for Nonlinear Wave Equations of Higher-Order Kirchhoff Type with Strong Dissipation
Guoguang Lin, Yunlong Gao, Yuting Sun
Department of Mathematics, Yunnan University, Kunming, China
Copyright © 2017 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: January 5, 2017; Accepted: February 10, 2017; Published: February 13, 2017
ABSTRACT
In this paper, we study on the initial-boundary value problem for nonlinear wave equations of higher-order Kirchhoff type with Strong Dissipation:. At first, we prove the existence and uniqueness of the local solution by the Banach contraction mapping principle. Then, by “Concavity” method 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.
Keywords:
Nonlinear Higher-Order Kirchhoff Type Equation, Strong Damping, Local Solutions, Blow-Up, Initial Energy
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
.
In this paper we shall deal with local existence and blow-up of solutions for nonlinear wave equations of higher-order Kirchhoff type with strong dissipation. The equation may be degenerate or nondenerate Kirchhoff equation, and derive the blow up properties of solutions of this problem with negative and positive initial energy by the method different from the references [5] - [13] .
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.
Cite this paper
Lin, G.G., Gao, Y.L. and Sun, Y.T. (2017) On Local Existence and Blow-Up of Solutions for Nonlinear Wave Equations of Higher-Order Kirchhoff Type with Strong Dissipation. International Journal of Modern Nonlinear Theory and Application, 6, 11-25. https://doi.org/10.4236/ijmnta.2017.61002
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