FINITE-TIME BLOW UP OF SOLUTIONS FOR A FOURTH-ORDER VISCOELASTIC WAVE EQUATION WITH DAMPING TERMS

In this paper, a class of fourth-order viscoelastic wave equations with damping terms is studied. First, the local existence and uniqueness of weak solutions for the proposed problem are proved by the linear approximation and the Faedo-Galerkin method. Next, a special case of the original problem is considered. Then, under some suitablely sufficient conditions on the relaxation functions and by using contrary arguments, we show that the corresponding problem in this case does not admit any global solutions. Ultimately, we prove the finite-time blow up of solutions in case of negative initial energy.


Introduction
In this paper, we consider the following initial-boundary value problem for a fourthorder viscoelastic equation u(x, 0) = ũ0 (x), u t (x, 0) = ũ1 (x), where µ, f, g, ũ0 , ũ1 , are given functions satisfying conditions specified later.In Eq. (1.1) 1 , the nonlinear term µ x, t, u, u x , u x (t) 2 depends on the integral u x (t) 2 = 1 0 u 2 x (x, t) dx.In view of structure, Eq. (1.1) is a very complex model and can be considered as a generalized one-dimensional model of fourth-order viscoelastic wave equations with Kirchhoff term.It is clear that the model in hand does not exist in the first place, however, of which more special forms have been mentioned in literature describing a variety of important physical processes.So we will introduce its development and evolution to show its background by listing several related models, For example, it concerns in some extensible beam models describing evolution of transverse deflection of an extensible beam obeying continuous dynamics, of which the original equation was proposed by Woinowsky-Krieger [28], and given by where β represents an initial axial displacement measured from the unstressed state and u(x, t) represents the transverse deflection of an extensible beam of natural length L whose ends are held a fixed distance apart.In [1], An studied a model related to elastoplastic-microstructure flows to explore the immediate post-critical behavior of the solutions For considering other wide applications in connection physics and mechanics related to some special cases of Eq. (1.1), we refer to nonlinear resonance of rectangular plates [7], or time-periodic transverse oscillations of a rod under external forces [8], or viscous flows in materials with memory [23] and the references therein.
After its appearance, numerous extensively interesting mathematical results of Eq. (1.5) such as local existence, global existence, asymptotic behavior and blow up in finite time of solutions have been investigated.As N = 1 (one-dimensional case), with substituting σ (u x ) x for β u 2 x x and adding the dissipative term u t , Yang [34] considered Eq.(1.5) in the form where σ(s) is a given nonlinear function and λ ≥ 0 is a real number.Then, if the initial energy is positive and suitably small, the global existence and asymptotic behavior of weak solutions are proved.Moreover, under some sufficient conditions on initial data and with the negative initial energy, the solution of Eq. (1.6) blows up in finite time.In [4], Chen and Lu considered Eq.(1.6) with replacing the weak dissipative term u t by the strong dissipation −u xxt and with the various boundary conditions.In the case that the nonlinear source f is a C 1 (R) function and f ′ (s) is bounded below, they proved the existence and uniqueness of the global generalized solution and the global classical solution, and also considered the finite time blow up with negative initial energy and the exponential decay with F (s) = s 0 f (τ )dτ ≥ 0.
Later, by using potential well method, Xu et al. [30] have extended and obtained the same results in [4] of finite time blow up with the initial energy E(0) satisfying 0 < E(0) < d (the depth of potential well).However, in their paper, the cases E(0) = d and E(0) > d were still an unsolved problem.In the frame of potential well theory, it seems that the initial energy must be less than the depth of potential well, so the high arbitrary initial energy case without such restriction becomes a very interesting and important problem.Subsequently, this unsolved problem has been solved and showed in some recent works, see for example [9] , [27], [35] (in onedimensional case), in which the authors established the finite time blow up of the solutions for the corresponding problem with arbitrary positive energy and suitable initial data.
For multi-dimensional forms, the following model of fourth-order equations named Kirchhoff plate equations and considered as a generalization of the Woinowsky-Krieger model to describe the large deflection of the plate [3], attract a lot of interest and are studied in various directions like local existence, global existence, nonexistence, stability and asymptotic behavior of solutions [11], [12], [15], [18], [30], [29], [33].Indeed, when α = 0, Lan et. al. [11] applied the Galerkin method and the weak compact method to Eq. (1.7) to obtain the existence and uniqueness of global solutions.Afterward, the same results of [11] have been extended by Long and Thuyet [18] in which the weakly nonlinear damping ε |u t | α−1 u t was replacing by the continuous and non-decreasing function g of Nemytsky-type operator.For Eq. (1.7) with weakly nonlinear damping and source, i.e. α = 0, M = 1, g = |u t | m−2 u t , f = |u| r−2 u, Ouaoua et al. [22] considered a nonlinear equation of Timoshenko type and then proved the local existence giving by the Faedo-Galerkin method, the global existence under suitable assumptions with positive initial energy, and the algebra stability of solution based on Komornik's integral inequality.Once again, the blow-up phenomena of solutions at arbitrary positive energy has been attracted a great deal of attention.In [29] , Wu and Tsai studied Eq. (1.7) in case of the absence of dispersions, i.e., α = 0 and g = 0, and established the blow-up properties with small positive initial energy after verifying the local existence by using the contraction mapping principle, and the global existence under some restrictions on the initial data.In the framework of potential well, considering Eq. (1.7) with nonlinear weak damping, linear strong damping and nonlinear exponential source, precisely as [33] have proved the blow-up phenomena of solutions with the high arbitrarily initial energy E(0) > 0 and the linear weak damping (r = 1).In addition, the local existence by using the contraction mapping principle, and the results of global existence, nonexistence and asymptotic behavior of solution for both subcritical initial energy level and critical initial energy level have also been discussed.Meanwhile, by exploiting the properties of the Nehari manifold and by constructing the appropriate and relaxed sufficient conditions, Liu et al. [15] used directly the relationship between the energy functionals associated with Eq. (1.7) in two-dimensional case and α = 0, g = u t , f = |u| p−1 u to obtain the global existence and finite-time blow up of solutions without the aid of d.
It is worth mentioning to the following abstract model of fourth-order equations including nonlinear strain and dissipative terms in the form in which a large amount of attention in various directions like well-posedness of weak solution, global existence, asymptotic analysis and blow up in finite time have been investigated.Actually, Esquivel-Avila [6] studied Eq. (1.8) with α = 0, , and established some sufficient conditions on the initial data to obtain the global solution and the finite time blow-up solution when E(0) ≤ d.Before that, Yang [34] has considered Eq.(1.8) with α = 0, g = u t , f = 0 and the strain term in more general form σ i = σ i (x i ), including the global existence and exponential decay with large initial data and small initial energy, and the finite time blow up of solutions in one-dimensional case N = 1.Recently, the results in [6], [16], [17], [34] were extended to that given in [12], [32], [33].Precisely, also in the structure frame of the potential well theory, the results of global existence, asymptotic behavior and blow up of solutions for both subcritical initial energy level and critical initial energy level have been proved; moreover, the finite-time blow up of solutions at arbitrarily positive initial energy has been discussed.There have been a large number of published papers on fourth-order equations that it is difficult to list all results; thus, we refer here more some recent models of fourthorder equations with various characteristic term such as [5] with nonlinear boundary damping and interior source, [15] with weak damping term and exponential source, [14] with Hardy-Hénon potential and polynomial nonlinearity, [21] with variableexponents, [31] with nonlinear damping and nonlinear source.
Considering that the above mentioned papers are devoted to the models with the single linear/nonlinear weak damping term or the single strong damping term sometimes the combinations of two of them; meanwhile few viscoelastic versions of the problem with a memory term, see [2], [19], [25], [26] and the references herein as examples; in which the authors got the existence of global attractor and asymptotic stability.Normally, the presence of memory term is one of factors causing decay property of solution energy for corresponding system; thus, problems with memory term seem difficult to obtain results of blow-up phenomena of soutions; in our observation, there seem have been no published results of finite-time blow up of solutions for fourth-order equations.In the present paper, we shall consider below a more general case with both the weak damping term and the strong damping term and the memory term, in which the finite-time blow up of solutions for the problem (1.1) in the case f = −λu t + f (u, u x ) and µ = µ u x (t) 2 u x + µ 1 (u, u x ) shall be studied.Precisely, we consider the initial-boundary value problem as follows where λ > 0 is a given constant and ũ0 , ũ1 , µ, µ 1 , g, f are given functions.Then, under suitable assumptions on initial data and some contrary arguments, we show that Prob.(1.9) does not admit any global solutions.Finally, by constructing the appropriate energy functionals and using arguments of continuity, we prove that the solution of Prob.(1.9) blows up in finite time.To best our knowledge, there seems no results of finite-time blow up of solutions for fourth-order viscoelastic problems such as Prob.(1.9) that includes both the weak damping term u t and the strong damping term −u xxt and the viscoelastic term

Preliminaries
Put Ω = (0, 1).Throughout this paper, we use the following notations: and respectively.Put ) Then, it is not difficult to prove the following lemmas (see [13]), hence the proofs of which are omitted the details.
Lemma 2.1.The imbedding Similarly, we also have the following lemma.

Lemma 2.4. The symmetric bilinear form a
There is a Hilbert orthonormal base {w j } ∞ j=1 of L 2 including the eigenfunctions corresponding to the eigenvalues λ j of the problem −∆w j = λ j w j in (0, 1) , w j (0) = w j (1) = 0, and satisfies The proof of Lemma 2.5 can be found in [ [24], p.87, Theorem 7.7], with H = L 2 , V = H 1 0 and u x , v x .Further, the Hilbert orthonormal base {w j } given in Lemma 2.5 can be explicitly determined by with respect to the scalar product a 1 (•, •) .
(ii) The sequence {w j /λ 2 j } is also a Hilbert orthonormal base of H 4 # with respect to the scalar product a 2 (•, •) .
On the other hand, w j satisfies the following boundary value problem:

Local existence and uniqueness
For a fixed T * > 0, we make the following assumptions: for all w ∈ H 2 ∩ H 1 0 , a.e., t ∈ (0, T ), together with the initial conditions where For each M > 0, we put For every T ∈ (0, T * ], we put which is a Banach space (see Lions [13]) with respect to the following norm with the corresponding norm For every M > 0, we consider two sets We construct a recurrent sequence {u m } defined by choosing the first term u 0 ≡ ũ0 , and suppose that then we find where The existence of u m is given by the following theorem.
Proof.The proof of Theorem 3.1 spends several steps as follows.
Step 1. Faedo-Galerkin approximation.Consider the basis {w j } for L 2 as in (2.4).Put where c (k) mj are determined by the following system of linear integro-differential equations in which The system (3.14)-(3.15) is equivalent to a system of linear intergal equations that can be rewritten in the following form where Using Banach's contraction principle, it is not difficult to prove that the fixed point equation (3.16) admits a unique solution c thus we omit the details of the proof.

Step 2. Priori estimates
Putting then it follows from (3.14) and (3.17) that Taking the derivative of F m (x, t) and µ m (x, t) (in (3.12)) up to second order and doing some calculations, we get the following lemma.
) where Using Lemma 3.1, and the following inequalities , we can estimate I 1 − I 5 on the right-hand side of (3.18) as follows ) For ds, we use integral by parts and estimate for I 6 as follows 2 ds, we estimate as follows.
Note that, we can choose T ∈ (0, T * ] such that and where Then, by (3.28) and (3.29), we obtain By using Gronwall's lemma, we deduce from (3.32) that for all t ∈ [0, T ] , for all m and k.This leads to the fact that ) , for all m and k. (3.34) Step 3. Limiting process.From (3.34), there exists a subsequence of {u Hence, Theorem 3.1 is proved completely.By Theorem 3.1, we prove the existence and uniqueness of weak solution of (1.1) which is given by the following theorem.Theorem 3.2.Let (A 1 ) − (A 4 ) hold.There are two postive constants M and T such that {u m } converges strongly in W 1 (T ) to u ∈ W 1 (M, T ) being the unique weak solution of (1.1).Moreover, the following estimate is claimed where k T ∈ (0, 1) is defined by (3.30), (3.31) and C T is a constant independent of m.
Proof.First, we shall prove that {u m } is a Cauchy sequence in W 1 (T ) .Let w m = u m+1 − u m .Then ūm satisfies where Using Lemma 3.2, and evaluating in a similar way as above, the terms J 1 − J 5 are estimated as follows Sm (s)ds, Sm (s)ds, (3.41) Sm (s)ds, It follows from (3.39), (3.41), that where D1 (M ), D2 are defined as in (3.31).
Further, by the assumptions (A 2 ) − (A 4 ), we obtain from (3.1) and (3.46) 5 that Thus, u ∈ W 1 (M, T ) .This confirms that u is a weak solution of (1.1).Next, we prove the uniqueness of weak solutions of (1.1) as follows. Let Taking w = v ′ (t) in (3.49) and integrating in t , we get where Then it follows from (3.50)-(3.51) that where M .By using Gronwall's lemma, we obtain that Z (t) ≡ 0, i.e., u = u 1 − u 2 = 0.This claims the uniqueness of solutions of (1.1 ).Thus, Theorem 3.2 is proved completely.

Blow-up of solutions
This section is devoted to studying of the finite-time blow up of the solution of the problem (1.1) in the case Precisely, we consider the initial-boundary value problem as follows where λ > 0 is a given constant and ũ0 , ũ1 , µ, µ 1 , g, f are given functions satisfiying the following assumptions By the same method used for the proof of Theorem 3.2, the problem (4.1) has a unique local solution provided by the following theorem.
In what follows, we will consider the existence of local solution of (4.1) in the case that the initial datum is less regularized.
Then, we also get the local solution existence given by the theorem below.
for a sufficiently small T > 0.
Proof.In order to obtain the existence of a weak solution, we use standard arguments of density. With For all m ∈ N, suppose that {ũ 0m } satisfies compatibility conditions Then, for each m ∈ N and the conditions of Theorem 4.1, there exists a unique function u m such that Priori estimates.Replacing v with u ′ m (t) and then integrating, we obtain where μ * = min{1, µ * , 2λ} and Similarly to the above estimates, we evaluate the terms I 1 − I 3 as follows Sm (s)ds, (4.10) Sm (s)ds, Sm (s)ds.
For the term I 4 , using the hypothesis ( Ã4 ), (ii), we get In order to evaluate u m (t) α L α .We use the estimate below In order to evaluate u mx (t) β L β .We use the estimates below and Then Thus, integrating in x from 0 to 1 for the above inequality, we have We deduce from (4.11), (4.12) and (4.13), the term I 4 is evaluated as follows By combining ( , we get where By the convergences given in (4.4), there exists a positive constant M 1 independent of m, such that S0m ≤ M 1 , for all m ∈ N. (4.17) It follows from (4.15) and (4.17) that Then, by solving the nonlinear Volterra integral equation (4.18) (based on the methods in [10]), we get the following lemma.

Lemma 4.1. There exists a positive constant T depending on T
where C T is a constant depending on T only.Next, we shall prove that {u m } is a Cauchy sequence in ) and then integrating with respect to t, we obtain where λ * = min{1, 2λ}, and We will evaluate the terms on the right-hand side of (4.21) as follows.Put Using the following inequalities w m,l x (t) 2 ≤ ∆w m,l (t) 2 , and we get estimates of Ī1 − Ī5 , as follows.
The term Ī2 .
The term Ī3 .From the below inequality we infer that Sm,l (s)ds.
The term Ī4 .From the below inequality we have Sm,l (s)ds.
The term Ī5 .Similarly, from the following inequality By Gronwall's lemma, we deduce from (4.30), that Hence This confirms that {u m } is a Cauchy sequence in W 1 (T ) .Then there exists u ∈ W 1 (T ) such that u m → u strongly in W 1 (T ) .(4.34) On the other hand, by (4.19), there exists a subsequence Moreover, by (4.23), we deduce that ) It follows from that (4.34), (4.36) that (4.37)By using the convergences (4.34), (4.35) and (4.37) to pass the limitations in (4.6), we have u ∈ W 1 (T ) satisfying the problem Finally, the uniqueness of a weak solution is obtained by using the well-known regularization procedure due to Lions, see Ngoc et.al. [20] as an example.
Theorem 4.2 is proved completely.
It is clear that Putting ) hold, then for any (ũ 0 , ũ1 ) ∈ H 2 ∩ H 1 0 × L 2 such that H(0) > 0, the weak solution u of (4.1) blows up in finite time, i.e., there is a positive constant T ∞ such that Proof.First, we prove that the problem (4.1) does not admit any global weak solutions. (4.40) Arguing by contradiction, we assume that the problem (4.1) admits a global weak solution

.41)
We define the energy functional associated with (4.1) by Put H(t) = −E(t), ∀t ≥ 0. Multiplying both sides of (4.1) 1 by u ′ (x, t) and then integrating the obtained equation, we have

.48)
In what follows, we show that there is a constant Multiplying both sides of (4.1) 1 by u(x, t) and integrating over [0, 1], it leads to (4.52) On the other hand, by (B 2 ) , (B 3 ) , we get Because of d 1 > p, hence we can choose δ 1 ∈ (0, 1) such that (1 − δ 1 )d 1 = p, then it follows from (4.52)  Remark 4.2.In some previous papers, the results of the finite time blowup for negative initial energy were derived in case of that the nonlinearity is less complex, for example such as in [4] and [30] with the nonlinearity f (u x ) x .Recently, with the same nonlinearity, the authors in [9] and [27] have proved that the blowup property occurs in finite time for arbitrary positive initial energy and suitable initial data.In our paper, the nonlinear quantities in the first equation of the problem (4.1) are considered with more complicated forms, precisely that are − ∂ ∂x µ u x (t) 2 u x + µ 1 (u, u x ) and f (u, u x ).Therefore, it is very difficult to construct sufficient conditions for which the finite-time blow up for positive initial energy of the solutions for the problem (4.1) is established, and hence this is still open.

Lemma 4 .
1 allows us to take T m = T for all m.