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On Local Existence and Blow-Up of Solutions for Nonlinear Wave Equations of Higher-Order Kirchhoff Type with Strong Dissipation

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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.

Cite this paper

*International Journal of Modern Nonlinear Theory and Application*,

**6**, 11-25. doi: 10.4236/ijmnta.2017.61002.

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