Global Well-Posedness for a Class of Kirchhoff-Type Wave System

In recent years, the small initial boundary value problem of the Kirchhoff-type wave system attracts many scholars’ attention. However, the big initial boundary value problem is also a topic of theoretical significance. In this paper, we devote oneself to the well-posedness of the Kirchhoff-type wave system under the big initial boundary conditions. Combining the potential well method with an improved convexmethod, we establish a criterion for thewell-posedness of the systemwith nonlinear source and dissipative and viscoelastic terms. Based on the criteria, the energy of the system is divided into different levels. For the subcritical case, we prove that there exist the global solutions when the initial value belongs to the stable set, while the finite time blow-up occurs when the initial value belongs to the unstable set. For the supercritical case, we show that the corresponding solution blows up in a finite time if the initial value satisfies some given conditions.

1.1.Historical Research.Kirchhoff-type wave system with nonlinear source and dissipative and viscoelastic terms have various applications in the field of physics and mechanics, which is the model to describe the motion of deformable solids.A single Kirchhoff-type wave equation is proposed: and (6) has its roots for the small amplitude vibrations of a string when  = 0 and  = 1, but the tension of the string can not be ignored (see, e.g., Carrier et al. [1]).While ( 6) is used to describe the dynamics of an elastic string with fading 2 Advances in Mathematical Physics memory when  ̸ = 0, this equation shows that the dynamic equilibrium of the object depends on both the present state of deformation and the history of the deformation gradient.Pohozaev and Tesei [2] proved that the solution exists in time if the datum satisfy an analytic-type condition for the case  = 0.This result of the case  ̸ = 0 was extended by Torrejon and Yong [3]; they obtained the existence of weakly asymptotic stable solution.Later, Munoz Rivera [4] showed the existence of global solutions for small initial value and the exponential decay of the total energy.Then Wu and Tsai [5] established the global existence and energy decay under the assumption   () ≤ −(), ℎ(  ) = −Δ  .Recently, this decay estimate was improved for a weaker condition on   () ≤ 0 in [6].
Problem (6) is simplified to the following format without viscoelastic term: in Ω × (0, +∞) , (7) and some results of (7) concerning global well-posedness have been established in [7][8][9][10][11] for the case of  ≡ 1.The above problem without source and dissipative terms is called Kirchhoff-type equation when  is not a constant function, which was first introduced by Kirchhoff [12]; it describes the nonlinear vibrations of an elastic string.up to now, there are numerous results related to global well-posedness, including global existence, decay result, and blow-up properties; we refer the reader to [13][14][15][16][17].
Most recently Xu and Yang considered the initial boundary value problem of the following equation in [18]; they gave a blow-up result under supercritical energy: Wave system such as (1) and (2) goes back to Reed [19] who proposed a similar system, but it does not contain () =  0 + (‖∇‖ 2 + ‖∇V‖ 2 )  and the viscoelastic terms ∫  0  1 (−)Δ , ∫  0  2 (−)Δ .Subsequently, concerning blow-up and nonexistence, results in wave systems were discussed.Agre and Rammaha [20] studied the following concrete system: in Ω ⊂ R  × (0, ) ( = 1, 2, 3) with initial and boundary conditions of Dirichlet type, where  1 (, V) and  2 (, V) satisfy (A1) and (A2).They obtained several results concerning global well-posedness of a weak solution and showed that any weak solution blows up in finite time at negative initial energy.Afterwards, Alives et al. made further efforts as regards (9) in [21].They obtained the global existence, uniform decay rates, and blow-up of solutions in finite time by involving the Nehari manifold when the initial energy is nonnegative and less than the mountain pass level value.
And this blow-up result was improved by Said-Houari [22] when the initial data are large enough.In [23], Rammaha and Sakuntasathien studied a more general case of (9) by degenerating damping terms.Several results on the existence of local and global solutions as well as uniqueness are obtained by considering the constraint on the parameters of the system.Furthermore, they proved that the weak solutions blow up in finite time whenever the initial energy is negative and the exponent of the source term is more dominant than the exponents of both damping terms.Moreover, many studies of the global well-posedness for wave systems with dissipative terms have been researched in [24][25][26][27][28].
Wave systems with viscoelastic terms and dissipative terms have not been fully studied.In [29] the following coupled nonlinear wave equations with dispersive terms, viscoelastic dissipative terms, and nonlinear weak damping terms are considered: A global nonexistence theorem for certain solutions with positive initial energy is proved.Reference [30] further considered a fourth-order wave system similar to (10).In that case, the energy increases exponentially when time goes to infinity and the initial data are large enough.
Recently [31] considered a system of two coupled wave equations with dispersive and strong dissipative terms under Dirichlet boundary conditions: where The global existence of weak solutions and uniform decay rates (exponential one) of the solution energy were established.
Many researches considered the initial boundary value problem with global existence and blow-up of solutions for the nonlinear wave equations as follows: in Ω × (0, +∞) , (13) where Ω is a bounded domain in R  ( = 1, 2, 3) with a smooth boundary Ω.When the viscoelastic terms   ( = 1, 2) are absent in (13), [20] showed local and global existences of a weak solution that any weak solution blows up in finite time with negative initial energy as the same way used in [22].Later, Said-Houari extended this blow-up result to positive initial energy.At the same time, Liu [32] studied the following Cauchy problem for the coupled system of nonlinear Klein-Gordon equations with damping terms: and the existence of standing wave with ground state was stated and a sharp criterion for global existence and blowup of solutions when (0) <  was established.The author introduced a family of potential wells and the invariant sets and vacuum isolating behavior of solutions for 0 < (0) <  and (0) ≤ 0, respectively.Furthermore, he proved the global existence and asymptotic behavior of solutions when the condition is 0 < (0) < .Finally, a blow-up result with arbitrarily positive initial energy was obtained.
Conversely, in the presence of the memory term (  ̸ = 0,  = 1, 2), some results related to the asymptotic behavior and blow-up of solutions of viscoelastic systems were discussed.For example, [33] studied problem (13) with (  ) = −Δ  and (V  ) = −ΔV  and obtained the fact that the decay rate of the energy function is exponential under suitable conditions on the functions   ,   ,  = 1, 2, certain initial data in the stable set.On the contrary, when the initial data is in the unstable set, the solutions blow up in finite time under positive initial energy.For (  ) = |  | −1   and (V  ) = |V  | −1 V  , [26] established local and global existence as well as finite time blow-up (the initial energy (0) < 0).The latter blowup result has been improved by [29] by considering a class of positive initial data.On the other hand, Messaoudi and Tatar [34] considered the following similar wave equations without considering coupling coefficient of Δ and ΔV: where the functions  1 and  2 satisfy the following assumptions: and for some constant  > 0 and 1 ≤   ≤ /( − 2),  = 1, 2, 3, 4, the solution goes to zero with an exponential or polynomial rate which depending on the decay rate of the relaxation functions   ,  = 1, 2 was obtained.This result was improved in [35] to weaker conditions on the relaxation functions and more general coupling functions.
Additionally, Liu and Wang [36] considered the following nonlinear hyperbolic systems with damping and source terms in Ω × (0, ] , The author defined potential well and the outer manifold of the potential well associated with system (17) and got the global existence in the case of (0) <  1 /2 and discussed the global nonexistence of solutions for problem (1)-( 5) in the case of (0) ≤ 0 and (0) <   on page 88 of [36].In [37], (1)-( 5) was considered with  ≡ 1 and without imposing the memory terms ( = ℎ = 0).The rate of decay of the exponential or polynomial energy of the damping terms was obtained.
(A2) () is a nonnegative  1 function for  ≥ 0 satisfying (A3) The nonlinearity of , ,  satisfies (A4) For  ≥ 0 the relaxation functions  1 and  2 are of class  1 and satisfy The solutions are global in time when the functions  1 ,  2 , and   ,  = 1, 2, satisfy suitable conditions and certain initial data is in the stable set.The author established the rate of decay of solutions by a difference inequality given by Nakao [39] and intended to study the blow-up phenomena of problem (1)-( 5).The blow-up of solutions when the energy is negative or subcritical case was proved by adopting and modifying the methods used in [29].In this way, the above results in [38] allowed a bigger region for the blow-up results and improved the results of Messaoudi and Said-Houari [29].More specifically, the decay result in [38] extends the one in [16,37] to problem (1)-( 5), where  is not a constant function and the equations considered in [38] have more dissipations.

Unsolved Problems.
It is well known that in the absence of the nonlinear source term the damping term ensures global existence.In addition, without the dissipative term, the nonlinear source term causes finite time blow-up of solution.Moreover, the viscoelastic materials possess a capacity of storage and dissipation of mechanical energy; therefore, it is interesting to investigate the well-posedness of solution for the viscoelastic equation with dissipative term and nonlinear source term.We can see that problem (1)-(5) contains system (9) ( ≡ 1,  1 =  2 = 0), systems (10), (13), and (15) ( ≡ 1), system (11) ( = 0,  ≡ 1,  1 =  2 = 0), and system (17) ( = 1,  ≡ 1,  1 =  2 = 0) as special cases.In fact, the viscoelastic terms change the frame of the equations comparing without those viscoelastic conditions, which makes the structure of the equations and solutions more complex and which also makes energy decay faster.So the classical methods can not be applied to investigate properties of solutions.Therefore, much less is known for (1)-( 5) with viscoelastic terms.We can note that in all of the above studies except [38] only the global well-posedness of solutions was proved in a relatively rough variational framework, but the global existence and finite time blow-up for problem (1)-(5) at the other initial energy levels have not been discussed yet.The global existence and blow-up results in [38] are under the assumption of negative initial energy (0) < 0 or (0) <  1 = (( − 1)/2( + 1))(1/( + 1)) 2/(−1) .In other words, to our knowledge there are no results on global well-posedness of solutions to the initial boundary value problem for a couple of nonlinear wave equations with coupling coefficient () =  0 + (‖∇‖ 2 + ‖∇V‖ 2 )  of Δ and ΔV, viscoelastic terms ∫  0  1 ( − )Δ , ∫  0  2 ( − )Δ , and nonlinear weak dissipative terms |  | −1   , |V  | −1 V  .It is natural to ask a question of how the solution behaves for problem (1)- (5), which is what we want to deal with in this paper.Moreover, regarding the initial energy level, the present paper is also a comprehensive study for low energy case and high energy situation.To our knowledge this is the first try to consider this problem.The most attractive one is that we attain a blow-up result with arbitrary positive initial energy for a wave system of Kirchhoff type.
We restricts our attention to considering the global existence and blow-up at two different initial energy levels.Since the initial energy level plays a crucial role in dealing with the well-posedness of problem ( 1)-( 5), the two cases are, respectively, tackled with different tools.For the subcritical case (0) < , there have been many tools tackling the hyperbolic problem without viscoelastic terms in [16,17].We may refer the tools to deal with (1)-( 5) with viscoelastic terms.By the well-known works [40][41][42][43][44], we see that the supercritical case (0) >  is not easy to deal with.Filippo and Marco [45] made the initial attempt to consider the global wellposedness of hyperbolic problem at high initial energy level   − Δ − Δ  +   = || −2 .However, they focused on particular source term || −1 , and our work is on the viscoelastic terms condition and the complex source term  1 (, V),  2 (, V).Based on the general comparison principle in [9], we try to resolve the above open problems with variational methods.In this paper, we consider the initial boundary value problem for system (10) with  =  = 0 and the nonlinear source terms, coupling coefficient (), the nonlinearity of , , , and the relaxation functions  1 and  2 satisfying the assumptions (A1)-(A4), respectively.In addition, we consider nonlinear damping terms of the form |  | −1   and |V  | −1 V  as in the first equation and in the second equation of (10), respectively.

The Main Results and Organization of the Paper.
In this paper, we mainly discuss the following problems.
(1) Case (0) < : different from the method applied in [38], we introduce a family of potential wells to obtain the results: invariant sets, global existence, and finite time blow-up.(2) Case (0) > : we obtain the finite time blow-up of solutions for problem (1)-( 5) whose initial data have arbitrarily high initial energy.
We can summarize our main conclusions in Table 1 and use the question mark "?" to indicate the open problem.The organization of the paper is as follows.
In Section 2, we introduce some notations, assumptions, and preliminaries.
From Sections 3-5, we prove the main results.

Notations and Primary Lemmas
In this section, we shall give some lemmas and some notations which will be used throughout this work.We use the standard Lebesgue space   (Ω) and Sobolev space  1 0 (Ω) with their usual norms and products as follows: We will use the embedding In this case, the embedding constant is denoted by  * ; that is, From assumption (A1) one can easily verify that Moreover we have the following result.Note that the following conclusion (Lemma 1) was assumed throughout many papers (see [16-31, 33, 34]); however in our opinion we think this conclusion is a deduction of assumption (A1).Thus we present this conclusion as follows and similar proof can be found in [29].
Lemma 1.There exist two positive constants  0 and  1 such that Proof.We can see that taking  1 = 2  + then the right-hand side of inequality ( 26) is trivial.For the left-hand side, the result is also trivial if  = V = 0. If, without loss of generality, For || ≤ |V|, we have ) .
Consider the continuous function So min () ≥ 0. If min () = 0 then, for some we have This implies that Consequently, and then If || ≥ |V|, similarly we have This leads to the desired result and completes the proof of Lemma 1.
As in [29], we still have the following results.
Remark 5 (see [46]).Condition (A1) is necessary to guarantee the hyperbolicity of the equation in ( 1) and ( 2) and condition ( 21) is needed to establish the local existence result.
Next for problem (1)-( 5) we introduce potential energy functional: Potential energy functional: Nehari functional: For the definition of (, V) please see assumption (A1) in the beginning of this paper.Moreover we introduce the potential well (stable set) and the outer space of potential well (unstable set) Moreover we define or equivalently where Lemma 6 (depth of potential well).The depth of potential well  = (( − 1) 1) , where  1 is defined in (26) and  * is the best imbedding constant from  1 0 (Ω) into  +1 (Ω).
Proof.From the definition of , we have (, V) ∈ N; that is, (, V) = 0. Then on the one hand from Lemma 1 we get Notice that, from assumption (A4) and the definitions  and  1 , we have that is On the other hand notice (, V) = 0; moreover by virtue of ( 38), (39), and (78), we get and hence we have 1) .
Lemma 7 (nonincreasing energy).Let (, V) be a solution of problem ( 1)-( 5); then () is a nonincreasing function for  ≥ 0; that is, Proof.Multiplying (1) by   and (2) by V  , integrating them over Ω, and then adding the results together and integrating by parts, it follows that Exploiting Lemma 3 on the third term and the fourth term on the right side of the above equality, we can obtain (48) for any regular solution.

Global Existence under the Case 𝐸(0) < 𝑑
Now we give the following definition of weak solution for problem (1)- (5).
Then we give the global existence of solutions for problem (1)-( 5) with low initial energy level (0) < .

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Definition 11 (finite time blow-up).A solution (, V) of problem ( 1)-( 5) is called a blow-up solution if there exists a finite time  such that lim sup By the same argument as Lemma 9, we can get the following lemma.
In order to prove Theorem 14 we state some relations of the depth of potential well , norm ‖∇‖ 2 + ‖∇V‖ 2 , and function (, V) as follows.

Lemma 13. Under the assumptions of Lemma 15, one has
Proof.From Lemma 6 for the depth of potential well , we have where  1 is defined in (26) and  * is the best imbedding constant from  1 0 (Ω) into  +1 (Ω).By Lemma 15, we get (, V) ∈ ; that is, (, V) < 0. Moreover by Lemma 1 and Sobolev embedding inequality, (, V) < 0 implies that Notice that, from assumption (A4) and the definitions  and  1 , we have that is . (78) Hence we obtain A finite time blow-up result of solutions for problem (1)-( 5) is showed as follows.
For the term  1 from we have Here by taking 2 1 =  + 1 and 2 2 =  + 1 and by (80) we have From Lemma 16 we can derive With the fact (0) <  we have So from (95) we have () >  1 > 0. Therefore we can derive Setting () = () −(−1)/4 , this inequality becomes This proves that () reaches 0 in finite time, say  →  * .Since  * is independent of the initial choice of  0 , we may assume that  * <  0 .This tells us that lim which completes the proof.

A Finite Time Blow-Up When 𝐸(0) > 0
We first present the following lemmas in order to prove Theorem 18.
Proof.Consider the following auxiliary ordinary differential equation: with the initial condition for every  ∈ [0,  0 ).
Moreover from (108) and (114) it implies that Suppose by contradiction that the first inequality of (113) is invalid; then there exists  1 ∈ [0,  0 ) such that From the continuity of the solution in time, there exists  0 ∈ [0,  0 ) such that On the other hand we have the following ordinary differential inequality for every  ∈ [0,  0 ).This ordinary differential inequality can be solved as which contradicts (118).Thus we prove the first inequality of (113), which together with (114) states (113).So we complete this proof.
Proof.Let  () = ‖‖ 2 + ‖V‖ 2 ; (123) then we have Adding ( 124) and (125) we have which, from the fact that (, V) ∈ , implies that Therefore, applying Lemma 15 with the fact that we can obtain that the map is strictly increasing.
In the following, we show the invariance of the unstable set under the flow of the problem (1)-( 5).