General Decay and Blow-Up of Solutions for a System of Viscoelastic Equations of Kirchhoff Type with Strong Damping

The general decay and blow-up of solutions for a system of viscoelastic equations of Kirchhoff type with strong damping is considered. We first establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy by exploiting the convexity technique, the other is for certain solutions with arbitrarily positive initial energy based on the method of Li and Tsai. Then, we give a decay result of global solutions by the perturbed energy method under a weaker assumption on the relaxation functions.

To motivate our work, let us recall some previous results regarding viscoelastic equations of Kirchhoff type.The following problem: 2 ) Δ + ∫  0  ( − ) Δ () d + ℎ (  ) =  () , (, ) ∈ Ω × (0, ∞) ,  = 0, (, ) ∈ Ω × [0, ∞) ,  (, 0) =  0 () ,   (, 0) =  1 () ,  ∈ Ω is a model to describe the motion of deformable solids as hereditary effect is incorporated.It was first studied by Torrejón and Yong [1] who proved the existence of a weakly asymptotic stable solution for large analytical datum.Later, Muñoz Rivera [2] showed the existence of global solutions for small datum and the total energy decays to zero exponentially under some restrictions.Then, Wu and Tsai [3] treated problem (2) for ℎ(  ) = −Δ  and proved the global existence, decay, and blow-up with suitable conditions on initial data.They obtained the blow-up properties of local solution with small positive initial energy by the direct method of [4].To obtain the decay result, they assumed that the nonnegative kernel   () ≤ −(), ∀ ≥ 0 for some  > 0. This energy decay result was recently improved by Wu  ( − ) Δ () d +  ()   + ||   = 0, (, ) ∈ Ω × (0, ∞) , with the same initial and boundary conditions as that of (2), and proved an exponential decay rate.This work extended the result of Zuazua [15], in which he considered (3) with  = 0 and the localized linear damping.By using the piecewise multipliers method, Cavalcanti and Oquendo [16] investigated the equation div [ ()  ( − ) ∇ ()] d +  () ℎ (  ) +  () = 0, (, ) ∈ Ω × (0, ∞) , with the same initial and boundary conditions as that of (2).Under the similar conditions on the relaxation function  as above, and () + () ≥  > 0 for all  ∈ Ω, they improved the results of [14] by establishing exponential stability for exponential decay function  and linear function ℎ, and polynomial stability for polynomial decay function  and nonlinear function ℎ, respectively.Concerning blow-up results, Messaoudi [17]  ( He proved that any weak solution with negative initial energy blows up in finite time if  >  and while exists globally for any initial data in the appropriate space if  ≥ .This result was improved by the same author in [18] for positive initial energy under suitable conditions on , , and .Recently, Liu [19] studied the equation ( − ) Δ () d − Δ  +   = || −2 , (, ) ∈ Ω × (0, ∞) , (7) with the same initial and boundary condition as that of (2).By virtue of convexity technique and supposing that where δ = max{0, }, he proved that the solution with nonpositive initial energy as well as positive initial energy blows up in finite time.
We should mention that the following system: 1 ( − ) Δ () d +           −1   =  1 (, V) , (, ) ∈ Ω × (0, ) , (, ) ∈ Ω × (0, ) ,  (, ) = 0, V (, ) = 0, (, ) ∈ Ω × [0, ) ,  (, 0) =  0 () ,   (, 0) =  1 () ,  ∈ Ω, was considered by Han and Wang in [20], where Ω is a bounded domain with smooth boundary Ω in R  ,  = 1, 2, 3.Under suitable assumptions on the functions   ,   ( = 1, 2), the initial data and the parameters in the above problem established local existence, global existence, and blow-up property (the initial energy (0) < 0).This latter blow-up result has been improved by Messaoudi and Said-Houari [21] into certain solutions with positive initial energy.Recently, Liang and Gao in [22] investigated the following problem: (, ) ∈ Ω × (0, ) , with the same initial and boundary conditions as that of (9).Under suitable assumptions on the functions   ,   ( = 1, 2) and certain initial data in the stable set, they proved that the decay rate of the solution energy is exponential.Conversely, for certain initial data in the unstable set, they proved that there are solutions with positive initial energy that blow up in finite time.It is also worth mentioning the work [23] in which we studied system (1).Under suitable assumptions on the functions   ,   ( = 1, 2) and certain initial conditions, we showed that the solutions are global in time and the energy decays exponentially.For other papers related to existence, uniform decay, and blow-up of solutions of nonlinear wave equations, we refer the reader to [14,[24][25][26][27][28][29] for existence and uniform decay, to [17,[30][31][32][33][34] for blow-up, and to [35][36][37][38][39][40] for the coupled system.To the best of our knowledge, the general decay and blow-up of solutions for systems of viscoelastic equations of Kirchhoff type with strong damping have not been well studied.Motivated by the above mentioned research, we consider in the present work the coupled system (1) with nonzero   ( = 1, 2) and nonconstant ().We note that in such a coupled system case we should overcome the additional difficulties brought by the treatment of the nonlinear coupled terms.We first establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy, the other is for certain solutions with arbitrarily positive initial energy.Then, we give a decay result of global solutions under a weaker assumption on the relaxation functions   () ( = 1, 2).
This paper is organized as follows.In the next section we present some assumptions, notations and known results and state the main results: Theorems 4, 5, 6, and 7.The two blowup results, Theorems 5 and 6, are proved in Sections 3 and 4, respectively.Section 5 is devoted to the proof of the decay result-Theorem 7.

Preliminaries and Main Result
In this section we present some assumptions, notations, and known results and state the main results.First, we make the following assumptions.
Remark 1.It is clear that () =  0 +  1   (which occurs physically in the study of vibrations of damped flexible space structures in a bounded domain in R  ) satisfies (A4) for  ≥ 0,  0 > 0,  1 ≥ 0,  ≥ 0 as long as  + 1 > .Indeed, by straightforward calculations, we obtain Next, we introduce some notations.Consider the Hilbert space  2 (Ω) endowed with the inner product and the functions  1 (, V) and  2 (, V) (see also [21]): where ,  > 0 are constants and  satisfies One can easily verify that where Then, we give two lemmas which will be used throughout this work.
Theorem 4. Suppose that (20), (1), and (2) hold, and that  0 , for some  > 0.Moreover, at least one of the following statements is valid: The energy associated with system (1) is given by 2 ) , for  ≥ 0. ( As in [5], we can get Then we have We introduce where  1 is the optimal constant in (24).Our first result is concerned with the blow-up for certain local solutions with nonpositive initial energy as well as positive initial energy.
To achieve general decay result we will use a Lyapunov type technique for some perturbation energy following the method introduced in [42].This result improves the one in Li et al. [23] in which only the exponential decay rates are considered.

Blow-Up of Solutions with Initial Data in the Unstable Set
In this section, we prove a finite time blow-up result for initial data in the unstable set.We need the following lemmas.
Then () > 0 for all  ∈ [0, ].Furthermore and, consequently, for almost every  ∈ [0,].Testing the first equation of system (1) with  and the second equation of system (1) with V, integrating the results over Ω, using integration by parts, and summing up, we have which implies Therefore, we have where Ψ(), () : [0, ] → R + are the functions defined by Using the Cauchy-Schwarz inequality, we obtain (the similar inequality for V () holds true) (the similar inequality for V ()) , Similarly, we have (the similar inequality for V () holds true)  (the similar inequality for V () holds true) (the similar inequality for V () holds true) .

Blow-Up of Solutions with Arbitrarily Positive Initial Energy
In this section, we prove the second blow-up result (Theorem 6) for solutions with arbitrarily positive initial energy.In order to attain our aim, we need the following three lemmas.
For the next lemma, we define Lemma 12. Assume that the conditions of Theorem 6 hold and let (, V) be a solution of (1), then Proof.By (86), we have Testing the first equation of system (1) with  and testing the second equation of system (1) with V and plugging the results into the expression of   () we obtain By using H ö lder's inequality and Young's inequality, we have Similarly, (93)  ) .(100) Let Then () satisfies (80) for  * = ( + 1)/2.By (81), we see that if that is, if which is satisfied by the second hypothesis to Theorem 6, then we get from Lemma 10 that   () > ‖∇ 0 ‖ 2 2 +‖∇V 0 ‖ 2 2 ,  > 0. Thus, the proof of Lemma 12 is completed.
In what follows, we find an estimate for the life span of () and prove Theorem 6.

General Decay of Solutions
In this section, we prove the general decay of solutions of system (1).The method of proof is similar to that of [42,Theorem 3.5].We first state a lemma which is similar to the one first proved by Vitillaro in [33] to study a class of a scalar wave equation.