Initial boundary value problem for a viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term: decay estimates and blow-up result

In this paper, we study the initial boundary value problem for the following viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term where the relaxation function satisfies g′(t)≤−ξ(t)gr(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g'(t)\leq -\xi (t)g^{r}(t)$\end{document}, t≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t\geq 0$\end{document}, 1≤r<32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\leq r< \frac{3}{2}$\end{document}. The main goal of this work is to study the global existence, general decay, and blow-up result. The global existence has been obtained by potential-well theory, the decay of solutions of energy has been established by introducing suitable energy and Lyapunov functionals, and a blow-up result has been obtained with negative initial energy.


Introduction
In this paper, we consider the following initial-boundary value problem with a delay term where ⊂ R n is a bounded domain with sufficiently smooth boundary ∂ .p ≥ 4, a, b, α, μ 1 are fixed positive constants, μ 2 is a real number, τ > 0 represents the time delay, and g is a positive function.
In the absence of the Balakrishnan-Taylor damping (α = 0), Problem (1.1) is reduced to the well-known nonlinear wave equation with b = g = 0 and a Kirchhof-type wave equation with g = 0, which has been extensively studied, see for instance [5,8,13,24,30,31,35,38,41,42] and the references therein.Balakrishnan-Taylor damping (α = 0), g = 0, and μ 1 = μ 2 = 0, was initially proposed by Balakrishnan and Taylor [2], and Bass and Zes [3].It is related to the panel flutter equation and to the spillover problem.So far, it has been studied by many authors, we refer the interested readers to [12,15,32,39,43,44] and the references therein.Zarai and Tatar [44] studied the following problem v tta + b ∇v 2  2 + σ ∇v∇v t dx v + t 0 h(ts) v(s) ds = 0. (1.2) They proved the global existence and the polynomial decay of the problem.Exponential decay and blow up of the solution to the problem were established in Tatar and Zarai [39].It is well known that time-delay effects often appear in many chemical, physical, and economical phenomena because these phenomena depend not only on the present state but also on the past history of the system.Nicaise and Pignotti [33] considered the following wave equation with a delay term They obtained some stability results in the case 0 < μ 2 < μ 1 .Then, they extended the result to the time-dependent delay case in the work of Nicaise and Pignotti [34].Kirane and Said-Houari [23] considered a viscoelastic wave equation with time delay They proved the global well posedness of solutions and established the decay rate of energy for 0 < μ 2 < μ 1 .Kafini et al. [17] investigated the following nonlinear wave equation with delay They proved the blow-up result of solutions with negative initial energy and p ≥ m, and we refer the interested readers to [9,10,18,27] and the references therein.For the viscoelastic wave equation with Balakrishnan-Taylor damping and time delay, Lee et al. [25] studied the following equation and established a general energy decay result by suitable Lyapunov functionals.Gheraibia et al. [14] considered the following equation and proved the general decay result of the solution in the case |μ 2 | < μ 1 .For the related works of PDEs with time delay, see for instance [6, 7, 11, 16, 19-22, 26, 28, 36, 37, 40] and the references therein.Motivated by the previous work, in this paper, we consider the problem (1.1) and under suitable assumptions on the relaxation functions g, we prove the global existence, general decay and the finite-time blow-up results of the solutions.
The outline of this paper is as follows: In Sect.2, we give some preliminary results.In Sect.3, we obtain the global existence of the solution of (1.1).Section 4 and Sect. 5 cover the general decay and blow-up of solutions, respectively.

Some preliminaries
In this section, we give some notation for function spaces and preliminary lemmas.Denote by • p and • H 1 to the usual L p ( ) norm and H 1 ( ) norm, respectively.
For the relaxation function g, we assume (A 1 ): (A 2 ): There exist a nonincreasing differentiable function ξ with ξ (0) > 0 satisfying Assume further that g satisfies By using direct calculations, we have where To deal with the time-delay term, motivated by Nicaise and Pignotti [33], we introduce a new variable which gives us Then, problem (1.1)is equivalent to Let ζ be a positive constant satisfying We first state a local existence theorem that can be established.
Next, we define the functionals and (2.17) Then, it is obvious that (2.18)

Global existence
In this section, we will prove that the global existence of the solution to (1.1) is in time.

General decay
In this section, we prove the general decay result by constructing a suitable Lyapunov functional.
Then, there exist two positive constants K and k such that the solution of problem (1.1) satisfies, for all ∀t ≥ t 0 , , r > 1. (4.2) ) For this goal, we set where ε is a positive constant to be specified later and In order to show our stability result, we need the following lemmas: Lemma 4.2 Let (v, z) be a solution of problem (2.8).Then, there exist two positive constants α 1 and α 2 such that for ε > 0 small enough.
Lemma 4.3 Assume that g satisfies (A 1 ) and (A 2 ), then

Corollary 4.4 ([4]
) Assume that g satisfies (A 1 ) and (A 2 ), and v is the solution of (1.1), then Lemma 4.5 Let (v, z) be a solution of problem (2.8).Then, the functional F(t) satisfies where k 1 and k 2 are some positive constants.

.17)
A simple integration of (4.17), leads to which implies

Case: 1 < r < 3/2
To establish (4.4), we note that from simple calculations show that (4.2) and (4.3) yield Next, let then, we have Applying Jensens's inequality for the second term on the right-hand side of (4.15) and using (A 2 ), we obtain Multiplying (4.26) by ξ ν (t)E ν (t), where ν = r -1, we have The remainder of the proof is similar to (4.2).The proof is complete.

Blow up
In this section, we state and prove the blow up of the solution to problem (1.1) with negative initial energy.Let where E(0) < 0. From (5.1) and (2.11) we have and H(t) is an increasing function.Using (2.10) and (5.1), we obtain Moreover, similar to the work of Messaoudi [29], we can obtain the following lemma that is needed later.
Theorem 5.2 Let the conditions of Lemma 5.1 hold.Then, the solution of problem (1.1) blows up in finite time. Proof where ε > 0 is a small constant that will be chosen later, and Taking a derivative of (5.4) and using the first equation in (2.8), we have Applying Hölder's and Young's inequalities, for η, δ > 0, we have ) and (5.9) Combining these estimates (5.7)-(5.9)and (5.6), we obtain εη(g • ∇v)(t).
solution of problem (1.1) blows up in finite time T * and T * ≤ Theorem 3.2 Assume that the conditions of Lemma 3.1 hold, then the solution (1.1) is global and bounded.
2. are some positive constants.