Existence, Decay, and Blow-Up of Solutions for a Higher-Order Kirchhoff-Type Equation with Delay Term

This article deals with the study of the higher-order Kirchhoff-type equation with delay term in a bounded domain with initial boundary conditions, where firstly, we prove the global existence result of the solution. Then, we discuss the decay of solutions by using Nakao’s technique and denote polynomially and exponentially. Furthermore, the blow-up result is established for negative initial energy under appropriate conditions.


Introduction
In this paper, we establish the higher-order Kirchhoff-type equation with delay term as follows: where A = −Δ, m ≥ 1is a natural number, q, r ≥ 0 are real numbers, p > 1 is a real number and is a bounded domain with smooth boundary ∂Ω in R n , n = 1 ; 2 ; 3; v is the outer normal. τ > 0 denotes time delay, and 1 and 2 are positive real numbers. The functions ðu 0 , u 1 , f 0 Þ are the initial data belong to a suitable space.
The problem (1) is a general form of a model introduced by Kirchhoff [1]. To be more precise, Kirchhoff recommended a model denoted by the equation for f = g = 0, for 0 < x < L, t ≥ 0, where uðx, tÞ is the lateral displacement, ρ is the mass density, h is the cross-section area, E is the Young modulus, L is the length, ρ 0 is the initial axial tension, and f , g are the external forces. Furthermore, (2) is called a degenerate equation when ρ 0 = 0 and nondegenerate one when ρ 0 > 0.
Time delays often appear in many various problems, such as thermal, economic phenomena, biological, chemical, and physical. Recently, the partial di/erential equations with time delay have become an active area, (see [2,3] and references therein). Datko et al. [4] indicated that a small delay in a boundary control is a source of instability. An arbitrarily small delay may destabilize a system which is uniformly asymptotically stable without delay unless additional conditions or control terms have been used in many cases [5]. Additional control terms will be necessary to stabilize hyperbolic systems including delay terms, (see [6][7][8] and references therein). In [6], Nicaise and Pignotti studied the equation as follows: where a 0 and a are positive real parameters. The authors obtained that, under the condition 0 ≤ α ≤ a 0 , the system is exponentially stable. In the case α ≥ α 0 , they obtained a sequence of delays that shows the solution is instable. In [8], Xu  In that work, the authors showed that an exponential stability result under the condition where d is a constant such that In recent years, some other authors investigate hyperbolic type equation with delay term (see [10][11][12][13][14][15][16]).
Without delay term ðμ 2 ju t ðx, t-τÞj r-1 u t ðx, t − τÞÞ, in 2004, Li [17] studied the higher-order Kirchho/-type equation as follows: where m > 1 is a positive integer, and q, p, r > 0 is a positive constant. The author obtained that the solution exists globally if p ≤ r, while if p > max fr, 2qg. He also established the blow-up result for E ð0Þ < 0. Later, in 2007, Messaoudi and Houari [18] obtained the blow-up of solutions with E ð 0Þ > 0 of the equation (6). Then, Piskin and Polat [19] considered global existence and decay estimates utilizing Nakao's inequality of the equation (6). Without delay term, when m = 1 and q = 0, equation (1) takes the form of a semilinear hyperbolic equation as follows: Georgiev and Todorova [20] obtained the blow-up of solutions for E ð0Þ < 0 if 1 < r < p ð1 < p < n/ðn − 2Þfor n ≥ 3, p > 1 for n < 3Þ of the equation (7). Under the condition of positive upper bounded initial energy, Vitillaro [21] proved the same results of equation (7). Also, Ohta [22,23] studied related problems for the blow-up results of the equation (7).
Messaoudi [24] studied the following equation and obtained an existence result for the equation (8) and proved that the solution continues to exists globally if r ≥ p; however, if r < p and the initial energy is negative, the solution blows up in finite time. Chen [25] established that the solution of (8) blows up with E ð0Þ > 0. In the presence of strong damping term ð−Δu t Þ, Piskin and Polat [26] obtained the decay estimates by using Nakao's inequality of equation (8).
When m = 1 and without delay term, equation (1) takes the form the following Kirchhoff-type equation: Many authors had studied existence and blow-up results at night time for equation (9) (see [27][28][29][30]). Ono [30] proved the blow-up results if p > max fr, 2γgðp < 2/ðn − 4Þfor n ≥ 5 , p > 0 for n ≤ 4Þ and E ð0Þ < 0 for equation (9). Later, Benaissa and Messaoudi [31] obtained the similar result for the generalized Kirchhoff-type equation as follows: where M : R + ⟶ R + and ϕðxÞ are bounded functions. Then, Wu [32], verified the same result of the general Kirchhoff-type equation with the positive upper bounded initial energy. In 2013, Ye [33] considered the global existence results by constructing a stable set in H 1 0 ðΩÞ and showed the decay by using a lemma of Komornik for the nonlinear Kirchhoff-type equation (11) with dissipative term. Moreover, Ye [34] obtained the global existence results by constructing a stable set in H m 0 ðΩÞ and showed the energy decay by using a lemma of V. Komornik for a nonlinear higher-order Kirchhoff-type equation with dissipative term is as follows: where A = ð−ΔÞ m , m > 1 is a positive integer. Gao et al. [35] considered the Kirchhoff-type equation without delay term as follows: The authors obtained the blow-up of solutions for E ð0Þ > 0 under appropriate conditions for equation (13).
In [36][37][38][39][40], some authors studied abstract evolution equations as follows: on suitable Banach space, and they proved some global 2 Journal of Function Spaces nonexistence of solutions. Some other authors studied related problems (see [41][42][43][44][45]). Motivated by the above works, we deal with the existence, decay, and blow-up results for the higher-order Kirchhoff type equation (1) with delay term and source term. There is no research, to our best knowledge, related to the higher-order Kirchhoff-type ðð Ð Ω jA m/2 uj 2 dxÞ q A m uÞ equation (1) with delay ðu 2 ju t ðx, t − τÞj r-1 u t ðx, t − τÞÞ and source ðjuj p−1 uÞ terms; hence, our work is the generalization of the above studies.
This work consists of five sections in addition to the introduction: Firstly, in Sect. 2, we recall some lemmas and assumptions. Then, in Section 3, we get the global existence of solutions. Moreover, in Section 4, we establish the decay results by using Nakao's tecnique. Finally, in Section 5, we obtain the blow-up of solutions for negative initial energy.

Preliminaries
In this part, we present some lemmas and assumptions for the proof of our result. Let H m ðΩ Þ denote the Sobolev space with the norm H m 0 ðΩ Þ denotes the closure in H m ðΩ Þ of C ∞ 0 ðΩÞ. For simplicity of notation, we denote by k⋅k p the Lebesgue space L p ð ΩÞ norm, k⋅k denotes L 2 ðΩÞ norm, and we write equivalent norm k∇⋅kinstead of H 1 0 ðΩÞ norm k⋅kH 1 0 ðΩÞ. We denote by C i ði = 1, 2, ⋯, nÞ various positive constants which may be different at different occurrences.
We make the assumptions on parameters r, p, and m as follows: (A1)

Global Existence
In this part, we consider the global existence results of the problem (1). Firstly, we introduce the new function z similar to the [7], Thus, we have Hence, problem (1) can be transformed as follows: 3

Journal of Function Spaces
We define the energy functional for any regular solution of (22) as follows: such that Also, have We easily see that Furthermore, we define Next, lemma gives that the energy functional E ðtÞ is a nonincreasing.
Lemma 3. Assume that ðu, zÞ is the solution of (22), then for t ≥ 0, Proof. We multiply the first equation in (22) by u t , integrate over, and use integration by parts, and we obtain d dt Integrating (30) over ð0, tÞ, we get We multiply the second equation in (22) by ςjzj r−1 z and integrate the result over Ω × ð0, 1Þ × ð0, tÞ, and we get By combining (31) and (32), we arrive at Utilizing the Young inequality on the fourth term of the left hand side of (33), we conclude that Deriving the (34), we have the desired result. Hence, the proof is completed. (24), we obtain

Remark 4. From the condition
Journal of Function Spaces Lemma 5. Assume that (19) and p > 2q + 1 hold. Let u 0 ∈ W and u 1 ∈ H m 0 ðΩÞ, such that then u ∈ W for each t ≥ 0.
Proof. It follows the continuity of uðtÞ, since Ið0Þ > 0, such that for some interval near t = 0. Assume that T m > 0 is a maximal time, when (26) holds on ½0, T m .
By (25) and (26), we obtain From (23), (38), and Lemma 3, we have Using Lemma 1 and (39), we get Thus, from (26), we arrive at IðtÞ > 0 for all t ∈ ½0, T m . T m is extended to T, by repeating the procedure. Hence, the proof is completed.
Proof. It is sufficient to show that ku t k 2 + kA m/2 uk 2ðq+1Þ is bounded independently of t. To obtain this, by using (23) and (26), we have since IðtÞ ≥ 0. Thus, where C = max f2, ð2ðq + 1Þðp + 1Þ/p − 2q − 1Þg. Therefore, we obtain the global existence of solutions. Therefore, we completed the proof.

Decay of Solution
In this part, we obtain the decay of solutions of the problem (22) by using Nakao's technique.

Blow-Up of Solution
In this part, we get the blow-up of solutions for negative initial energy, in the case r > 1.

Conclusions
Time delays often appear in many various problems, such as, thermal, economic phenomena, biological, chemical, and physical. Recently, the partial differential equations with time delay have become an active area (see [2,3] and references therein). In recent years, there has been published much work concerning the wave equation with constant delay or time-varying delay. However, to the best of our knowledge, there were no global existence, decay, and blow-up results for the higher-order Kirchhoff-type equation with delay term. Firstly, we have been obtained the global existence result. Later, we have been established the decay results by using Nakao's technique. Finally, we have proved the blow-up of solutions with negative initial energy for the problem (1) under the sufficient conditions in a bounded domain. In the next work, we will extend our current study to more general case of the problem (1).