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 [1] 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 [2] [3] [4] .

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 [5] .

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 [6] .

When, reference [7] 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 [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] .

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

Lemma 2.2 [9] Suppose that and is a nonnegative function such that

If

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

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

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

where.

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

where

and

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

where.

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

where

and.

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.

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

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

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

have for;

4) If and

hold, then we get for.

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

and

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

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

So

Thus, we have

where

Set

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

where is some certain constant which will be chosen later. Then we get

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, Multiplying (4.23) by and integrating it from to, we have

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

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 [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

.

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.

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