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.
Nonlinear Higher-Order Kirchhoff Type Equation Strong Damping Local Solutions Blow-Up Initial Energy1. 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:
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  first investigated the following nonlinear vib- ration of an elastic string for:
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:
It has been extensively studied and several results concerning existence and blowing-up have been established    .
When, the Equation (1.1) becomes the following Kirchhoff equation with Lipschitz type continuous coefficient and strong damping:
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  .
When, the Equation (1.1) becomes the following Kirchhoff equation:
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  .
When, reference  has considered global existence and decay esti- mates for nonlinear Kirchhoff-type equation:
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  -  .
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  .
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  ) Let be a number with
and. Then there is a constant
depending on and such that
Lemma 2.2  Suppose that and is a nonnegative function such that
then we have. Here, is a constant and
the smallest positive root of the equation
Lemma 2.3  If is a non-increasing function on such that
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
Proof. We proof the theorem by Banach contraction mapping principle. For and, we define the following two-parameter space of solutions:
where. Then is a complete metric space with the distance
We define the non-linear mapping in the following way. For is the unique solution of the following equation:
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
To proceed the estimation,we observe that for. By Lemma 2.1, we have
Because of (if), then
Since by the Young inequality, we see that
Combining these inequalities, we get
Therefore, by the Gronwall inequality, we obtain
So, for all, we obtain
Therefore, in order that the map verifies 1), it will be enough that the parameters and satisfy
Moreover, it follows from (3.14) that and. It implies
Next, we prove 2). Suppose that (3.15) holds. We take, let, and set. Then satisfies
Multiplying (3.17-3.18) by and integrating it over and using Green’s formula, we have
To proceed the estimation, by Lemma 2.1 observe that
Substituting (3.22)-(3.24) into (3.21), we obtain
According to the same method, Multiplying (3.17-3.18) by and inte- grating it over, we get
Taking (3.25) (3.26) and by (3.10), it follows that
Applying the Gronwall inequality, we have
So, by (3.10) we have
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
Then, we have
Definition 4.1 A solution of (1.1)-(1.3) is called a blow-up solution, if there exists a finite time such that
For the next lemma, we define
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
4) If and
hold, then we get for.
Proof. Step 1: From (4.4), we obtain
From the above equation and the energy identity and, we obtain
Therefore, we obtain 1).
Step 2: If, then by (i), we have
Integrating (4.8) over, we have that
Thus, we get for, where
So, 2) has been proved.
Step 3: If, then for we have
Integrating (4.10) over, we have that
And because of, then we get
Thus, 3) has been proved.
Step 4: For the case that, we first note that
By using Hölder inequality, we have
Thus, we have
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:
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:
Then the solution blow up at finite. And can be estimated by (4.26)-(4.29), respectively, according to the sign of.
where is some certain constant which will be chosen later. Then we get
By the Hölder inequality, we obtain
By 1) of Lemma 4.1, we get
Then, we obtain
Therefore, we get
Note that by Lemma 4.2, Multiplying (4.23) by and integrating it from to, we have
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
Next, are estimated respectively according to the sign of and Lemma 2.3.
In case 1), we have
Furthermore, if, then we have
In case 2), we get
In case 3), we obtain
where. Note that in case 1), is given Lemma 4.1, and in
case 2) and case 3).
Remark 4.1  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,
if, is chosen to satisfy, where, Therefore, we have
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.
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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