Blow up and asymptotic behavior for a system of viscoelastic wave equations of Kirchhoff type with a delay term

The focus of the current paper is to investigate the initial boundary value problem for a system of viscoelastic wave equations of Kirchhoff type with a delay term in a bounded domain. At first, the energy decay rate is proved by Nakao’s technique and expressed polynomially and exponentially depending on the parameter m. However and in the unstable set, for certain initial data, the blow up of solutions is obtained.

This type of problem without delay (i.e., µ 2 = 0) has been considered by many authors during the past decades and many results have been obtained (see [9], [18], [22], [24] ) and the references therein.
The problem (1.1)-(1.4) without the viscoelastic term and delay (i.e., g = 0, µ 2 = 0) has been extensively studied and decay and blow-up have been established. For example, the following equation has been considered by Matsuyama and Ikehata in [18], for g(u t ) = δ|u t | p u t and f (u) = ξ|u| p u. The authors proved existence of the global solutions by using Faedo-Galerkin's method and the decay of energy based on the method of Nakao [20]. Later, Ono [21] investigated equation (1.5) for M (s) = bs, f (u) = ξ|u| p u and g(u t ) = −∆u t . They showed that the solutions blow up in finite time with E(0) ≤ 0. For M (s) = a + bs and g(u t ) = u t , this model was considered by the same author in [22]. By applying the potential well method, he obtained the blow-up properties with positive initial energy E(0). Recently, Zeng et al. [24] have studied equation (1.5) for the case g(u t ) = u t with initial condition and zero Dirichlet boundary condition. By using the concavity argument, they proved that the solutions to equation (1.5) blows up in finite time with arbitrarily high energy. When g = 0 and M is not a constant function, problems related to (1.5) have been treated by many authors. Wu and Tsai [23] considered the global existence, asymptotic behavior and blow-up properties for the following equation where (x, t) ∈ Ω × (0, ∞) and with the same initial and boundary conditions as that of problem (1.1)- (1.4). To obtain the decay result, they assumed that the nonnegative kernel g (t) ≤ −rg(t) ∀t ≥ 0 for some r > 0. Later, Wu [25], extended the result of [23] under a weaker condition on g (i.e g (s) ≤ 0 for t ≥ 0).
In the present paper, we analyze the influence of the viscoelastic, damping and delay terms on the solutions to (1.1)- (1.4). Under suitable assumptions on the function g, the initial data and the parameters in the equations, we establish several results concerning asymptotic behavior and finite blow-up of solutions to (1.1)-(1.4) for both negative and positive initial energy.
The paper is organized as follows. In Section 2, we present the preliminaries and some lemmas. In Section 3, the decay property is discussed. Finally, in Section 4, the blow-up results of (1.1)-(1.4) are obtained on different cases of the sign of the initial energy E(0).

Preliminary Results
In this section, we present some material for the proof of our result. We assume where M (s) = s 0 M (ν)dν, m 1 > 0, β and η are positive constants that will be specified later.
where c 2 s is the Poincaré constant and l is given in (A 1 ).

Global existence and asymptotic behavior
In order to prove the global existence result, we introduce the new variable z as in [12], Therefore, problem (1.1)-(1.4) can be transformed as follows For any regular solution of (3.1), we define the energy as Lemma 3.1. Let (u, z) be the solution of (3.1), then the energy satisfies Proof. Multiplying the first equation in (3.1) by u t , integrating over Ω and using integration by parts, we get d dt (3.5) Using Lemma 2.3 on the last term of the left hand side of (3.5), we find d dt Integrating (3.6) over (0, t) we arrive at (3.7) Multiplying the second equation in (3.1) by ζ|z| m−1 z and integrating the result over Ω × (0, 1) we obtain (3.8) Combining (3.7) and (3.8) together, we get Making use of the Young's inequality on the fourth term of the left hand side of (3.9), we deduce that   We show that when m ≥ p the solution of the problem (3.1) is global if Now, we center our attention on the global existence of the solutions to the problem (3.1). In order to do so, as in [6], we define and (3.14) We observe that and and (3.11) hold. Let (u, z) be the solution of the problem (3.1). Assume further that I 1 (0) > 0 and Proof. Since I(0) > 0, then there exists (by continuity of u(t)) T * < T such that From (3.14) and (3.15) we get easily Thus by (3.10) and (3. Exploiting Lemma 2.1 and formula (3.17), we obtain (3.21) Whereupon Repeating this procedure and using the fact that We can take T * = T . This completes the proof.
(Ω) and f 0 ∈ L 2 (Ω × (0, 1)) be given. Suppose that (3.11) and µ 2 < µ 1 hold. Then the solution of the problem (3.1) is global and bounded in time. Furthermore, if then we have the following decay estimates: Proof. First, we prove that T = ∞. It is sufficient to show that l ∇u 2 2 is bounded independently of t. We have from (3.14) and (3.15) (0), where ξ is a positive constant which depends only on p, thus we obtain the global existence result. This ends the proof.
Moreover, integrating (3.42) over (t, t 2 ), using (3.24) and taking into account the fact that E(t 2 ) ≤ 2 which implies that Then a simple application of Young's inequality gives, for all t ≥ 0 where c 11 , c 12 are some positive constants. Therefore, we have the following decay estimate: here we choose c 12 > 1. Thus by Lemma 2.2, we obtain with τ 1 = ln c 12 c 12 −1 .

Blow up result
In this section we have a condition on parameters m and p as follows For this case our result reads as follows: (Ω) and f 0 ∈ L 2 (Ω × (0, 1)) be given. Assume that the assumptions (A 0 ) − (A 2 ) are fulfilled. Let p > 2 and suppose that (4.1) holds. Then for any initial data satisfying E(0) < 0, the solution of (3.1) blows up in finite time.
Case 2: if ϕ(t * ) ≤ λ 1 , then by continuity of the function ϕ(t), there exists 0 < t 1 < t * such that This is also a contradiction of the first estimate of (4.36). Thus, we have proved the first estimate in (4.36).
To prove the second estimate in (4.36) we have Then from (4.38) we have (4.40) This completes the proof.