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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.

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

It is known that Kirchhoff [

where

When

It has been extensively studied and several results concerning existence and blowing-up have been established [

When

where

When

where

When

where

outward normal 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

In this section, we introduce material needed in the proof our main result. We use the standard Lebesgue space

abbreviations:

Lemma 2.1 (Sobolev-Poincaré inequality [

depending on

Lemma 2.2 [

If

then we have

Lemma 2.3 [

where

Moreover, for the case that

If

If

Theorem 3.1 Suppose that

for any given

Proof. We proof the theorem by Banach contraction mapping principle. For

where

We define the non-linear mapping

We shall show that there exist

1)

2)

First, we shall check (i). Multiplying Equation (3.4) by

integrating it over

where

To proceed the estimation,we observe that for

Because of

Since

Combining these inequalities, we get

Therefore, by the Gronwall inequality, we obtain

where

and

So, for all

Therefore, in order that the map

Moreover, it follows from (3.14) that

Next, we prove 2). Suppose that (3.15) holds. We take

Multiplying (3.17-3.18) by

To proceed the estimation, by Lemma 2.1 observe that

where

Substituting (3.22)-(3.24) into (3.21), we obtain

According to the same method, Multiplying (3.17-3.18) by

Taking (3.25)

where

and

Applying the Gronwall inequality, we have

So, by (3.10) we have

where

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

Then, we have

where

Definition 4.1 A solution

For the next lemma, we define

Lemma 4.1 Suppose that

1)

2) If

3) If

have

4) If

hold, then we get

Proof. Step 1: From (4.4), we obtain

and

From the above equation and the energy identity and

(4.7)

Therefore, we obtain 1).

Step 2: If

Integrating (4.8) over

Thus, we get

So, 2) has been proved.

Step 3: If

Integrating (4.10) over

And because of

Thus, 3) has been proved.

Step 4: For the case that

By using Hölder inequality, we have

So

Thus, we have

where

Set

Then

and Lemma 2.2, then

Lemma 4.2 Suppose that

1)

2)

3)

hold.

Then, there exists

Proof. By Lemma 4.1,

Theorem 4.1 Suppose that

1)

2)

3)

hold.

Then the solution

Proof. Let

where

and

where

By the Hölder inequality, we obtain

where

By 1) of Lemma 4.1, we get

Then, we obtain

Therefore, we get

Note that by Lemma 4.2,

where

When

we also have

Then by Lemma 2.3, there exists a finite time

and the upper bounds of

Next,

In case 1), we have

Furthermore, if

In case 2), we get

In case 3), we obtain

where

case 2) and case 3)

Remark 4.1 [

1) In the case

2) In the case

3) For the case

here

if

In this paper, we prove that nonlinear wave equations of higher-order Kirchhoff Type with Strong Dissipation exist unique local solution on

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.

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