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

In this paper, we study on the initial-boundary value problem for nonlinear wave equations of higher-order Kirchhoff type with Strong Dissipation: ( ) ( ) ( ) 2q p m m m tt t u u a b D u u u u + −∆ + + −∆ = . 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): ( ) 0 0 E < , in the case 2): ( ) 0 0 E = and in the case 3): ( ) 0 0 E > . At last, we consider that the estimation of the upper bounds of the blow-up time T ∗− is given for deferent initial energy.

, 0 , , 0 , , where Ω is a bounded domain in n R with the smooth boundary ∂Ω and v is the unit outward normal on ∂Ω .Moreover, 1 m > is an integer constant, and q , p , a and b are some constants such that 1 q ≥ , 0 p ≥ , 0 a ≥ , 0 b ≥ and 0 a b + > .We call Equation (1.1) a non-degenerate equation when 0 a > and 0 b > , and a degenerate one when 0 a = and 0 b > .In the case of 0 a > and 0 b = , Equation (1.1) is usual semilinear wave equations.It is known that Kirchhoff [1] first investigated the following nonlinear vibration of an elastic string for 0 f δ = = : ; 0 , 0, 2 where ( ) is the lateral displacement at the space coordinate x and the time t ; ρ : the mass density; h : the cross-section area; L : the length; E : the Young modulus; 0 p : the initial axial tension; δ : the resistance modulus; and f : the external force.( ) [ ) , , 0, , , 0 , , 0 , , ≥ is a bounded domain with a smooth boundary ∂Ω .p > 2 and It has been studied and several results concerning existence and blowing-up have been established [5].
When 1 m = , the Equation (1.1) becomes the following Kirchhoff equation: where Ω is a bounded domain in n R with the smooth boundary ∂Ω and v is the unit outward normal on ∂Ω .Moreover, q , p , a and b are some constants such that 1 q ≥ , 0 p ≥ , 0 a ≥ , 0 b ≥ and 0 a b + > .It has been studied and several results concerning existence and blowing-up have been established [6].
When 1 m = , reference [7] has considered global existence and decay esti- mates for nonlinear Kirchhoff-type equation: , , 0, , where Ω is a bounded domain of ( ) 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 estimate for blow-up time T * by lemma of [9].

Preliminaries
In this section, we introduce material needed in the proof our main result.We  and introduce the following abbreviations: . Then there is a constant K depending on Ω and s such that , .
Lemma 2.2 [9] Suppose that 0 δ > and ( ) Moreover, for the case that an upper bound of
Proof.We proof the theorem by Banach contraction mapping principle.For 0 T > and 0 R > , we define the following two-parameter space of solutions: where ( ) ( ) We define the non-linear mapping S in the following way.For , , is the unique solution of the following equation: We shall show that there exist 0 T > and 0 R > such that 1) S maps 2) S is a contraction mapping with respect to the metric ( ) First, we shall check (i).Multiplying Equation (3.4) by ( ) , and integrating it over Ω , we have To proceed the estimation,we observe that for where Therefore, in order that the map S verifies 1), it will be enough that the parameters T and R satisfy ( ) ( ) .
Next, we prove 2).Suppose that (3.15) holds.We take 1 2 , u Sv u Sv = = , and set , 0 0, , 0 0, .Multiplying (3.17-3.18)by 2 t w and integrating it over Ω and using Green's formula, we have To proceed the estimation, by Lemma 2.1 observe that ( ) where 0 1 ) ( ) According to the same method, Multiplying (3.17-3.18)by 2Aw and inte- grating it over Ω , we get Taking (3.25) 1 3 + × (3.26) and by (3.10), it follows that ( ) ( ) 2 72 72 , q q q q p p q q q q p p e w t Applying the Gronwall inequality, we have can see S is a contraction mapping.Finally, we choose suitable R is suffi- ciently large and T is sufficiently small, such that 1) and 2) hold.By applying Banach fixed point theorem, we obtain the local existence.

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 u of (1.1)-(1.3)by ( ) Then, we have where ( ) 3) is called a blow-up solution, if there exists a finite time T * such that ) and hold.Then we have the following results, which are 1) hold, then we get ( ) Step 1: From (4.4), we obtain From the above equation and the energy identity and 2 p q ≥ , we obtain ( Therefore, we obtain 1).
Step 4: For the case that ( ) By using Hölder inequality, we have ( ) and Lemma 2.2, then ( ) ) and hold and that eigher one of the following conditions is satisfied: Then, there exists 0 0 t ≥ , such that ( ) Proof.By Lemma 4.1, 0 t t * = in case (i) and 0 0 t = in case 2) and 3).
) and hold and that eigher one of the following conditions is satisfied: 2) ( ) Then the solution u blow up at finite T * .And T * can be estimated by (4.26)-(4.29),respectively, according to the sign of ( ) where 1 T is some certain constant which will be chosen later.
we also have 0 Then by Lemma 2.3, there exists a finite time T * such that ( ) and the upper bounds of T * are estimated respectively according to the sign of ( ) . This will imply that In case 1), we have ( ) ( ) , then we have Remark 4.1 [10] The choice of 1 T in (4.17) is possible under some conditions.

Conclusion
In this paper, we prove that nonlinear wave equations of higher-order Kirchhoff Then, we establish three blow-up results for certain solutions in the case 1): ( ) 0 0 E < , in the case 2): ( ) 0 0 E = and in the case 3): ( ) 0 0 E > .At last, we consider that the estimation of the upper bounds of the blow-up time T * − is given for deferent initial energy.
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 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- Equation (1.1) becomes a nonlinear wave equation:

7 )
It has been extensively studied and several results concerning existence and blowing-up have been established[2] [3Equation (1.1) becomes the following Kirchhoff equation with Lipschitz type continuous coefficient and strong damping: 22)-(3.24) into (3.21),we obtain measures, and v is the unit outward normal on ∂Ω , and v 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