General Decay of a Nonlinear Viscoelastic Wave Equation with Balakrishnân-Taylor Damping and a Delay Involving Variable Exponents

This paper was aimed at investigating the stability of the following viscoelastic problem with Balakrishnân-Taylor damping and variable-exponent nonlinear time delay term u tt t − τ Þ = 0in Ω × ℝ + , where Ω is a bounded domain of ℝ n , p ð : Þ : (cid:1) Ω ⟶ ℝ is a measurable function, g > 0 is a memory kernel that decays exponentially, α ≥ 0 is the potential, and M ðk ∇ u k 22 Þ = a + b k ∇ u ð t Þk 22 + σ Ð Ω ∇ u ∇ u t dx for some constants a > 0 , b ≥ 0 , and σ > 0 . Under some assumptions on the relaxation function, we use some suitable Lyapunov functionals to derive the general decay estimate for the energy. The problem considered is novel and meaningful because of the presence of the ﬂ utter panel equation and the spillover problem including memory and variable-exponent time delay control. Our result generalizes and improves previous conclusion in the literature.


Introduction
In recent years, much attention has been paid to the study systems with variable exponents of nonlinearities which are models of hyperbolic, parabolic, and elliptic equations. These models may be nonlinear over the gradient of unknown solutions and have nonlinear variable exponents. Researches of these systems usually use the imbedding of Lebesgue and Sobolev spaces with variable exponents (see, e.g., [1,2]). Or see [3][4][5][6][7][8][9][10][11][12][13][14] and the references therein for more details of relevant problems.
In this paper, we concentrate on the asymptotic behavior of weak solutions for the following weakly damped viscoelastic wave equation with Balakrishnân-Taylor damping and variable-exponent nonlinear time delay term where u : Ω × ½0,∞Þ ⟶ ℝ is unknown function, μ 1 ≥ 0,μ 2 is a real number, τ > 0 is the time delay, g > 0 is a memory kernel, and α ≥ 0 is the potential.
Much attention has been paid to the simulation of phenomena such as the vibration of elastic strings and elastic plates, when g = 0, and μ 1 = μ 2 = 0; equation (1)) degrades into the Kirchhoff's original equation which was first introduced to study the oscillations of stretched strings and plates in [15]. In addition, equation (2) is also said to be the wave equation of Kirchhoff type, where the unknown function u = uðx, tÞ represents lateral deflection and E, ρ, h, L, p 0 , and f , respectively, denote Young's modulus, mass density, cross-section area, length, initial axial tension, and external force. The Kirchhoff equation has been investigated in a lot of articles due to its abundant physical background. At the present paper, we try to mention some considerable efforts on this topic. There are many important results, such as the local solutions in time, well-posedness, and solvability; for the Kirchhoff type, equation (2) in general dimensions and domains has been obtained in lots of articles (see, e.g., [16][17][18][19][20][21][22][23][24] and the references therein).
On damping terms, we point out several excellent works: Lian and Xu in [33] studied a class of nonlinear wave equations with weak and strong damping terms, and they established the existence of weak solutions and related blow-up results under three different initial energy levels and different conditions. Yang et al. [34] investigated the exponential stability of a system with locally distributed damping. Lian et al. [35] were interested in a fourth-order wave equation with strong and weak damping terms; they obtained the local solution, the global existence, asymptotic behavior, and blow-up of solutions under some condition.
Time delays are common phenomena in many physical, chemical, biological, thermal, and so on (see [36][37][38] for more details). Several authors have investigated existence and stability of the solutions to the viscoelastic wave equation involving delay term under some appropriate conditions on μ 1 , μ 2 , and g (see, e.g., [39]). For other related problems, one can also refer to [40][41][42][43][44]. The terminology variable exponents mean that pð:Þ is a measurable function and not a constant. This term μ 1 ju t j pð:Þ−2 u t + μ 2 ju t ðt − τÞj pð:Þ−2 u t ðt − τÞ is a generalization of μ 1 u t + μ 2 u t ðt − τÞ, which corresponds to pð:Þ > 1. In fact, (1) is also an extension of the second-order viscoelastic wave equation under variable growth conditions which is obtained when considering μ 1 ju t j pð:Þ−2 u t + μ 2 ju t ðt − τÞj pð:Þ−2 u t ðt − τÞ: Equation (3) is a well-known electrorheological fluid model that appears in fluid dynamic treatment (see in [45]). However, the researches related to the viscoelastic wave equation possessing delay terms, Balakrishnân-Taylor damping, and variable growth conditions are not sufficient, and the results about these equations are relatively rare (see [46]). In particular, in [40], the authors considered this class of equations under some suitable assumptions; they use suitable Lyapunov functionals to derive general energy decay results, and one see similar work in [44]. Mingione and Rădulescu [47] were concerned with the regularity theory of elliptic variational problems under nonstandard growth conditions. This paper devotes to generalize some previous results. In particular, in this case, we will use the relaxation function, the specified initial data, and a special Lyapunov functional, which depends on the behavior of the relation function and is not necessary to decay in some polynomial or exponential form, to get a general decay estimate of the energy.
In addition to the introduction, this paper is divided into two parts. In Section 2, we review some basic definitions about Lebesgue and Sobolev spaces with variable exponentials and give some related properties. At the end of this section, we present our main results. In Section 3, we prove our results, showing that a solution of (1) possesses a general decay with small initial values ðu 0 , u 1 Þ.

Functional Setting and Main Results
In this section, we will give some preliminaries and our main results.
Without loss of generality, hereinafter, we suppose Ω ⊆ ℝ n (n ≥ 1) is a bounded domain with smooth boundary Γ. Moreover, let p : Ω ⟶ ð1,+∞Þ be a measurable function and denote As in [1,48,49], we define the following variableexponent Lebesgue spaces and Sobolev spaces. The first one is the variable-exponent space L pð:Þ ðΩÞ: and it is obvious a Banach space with the following 2

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Luxemburg norm Actually, in many respects, variable-exponent Lebesgue spaces are very similar to classical Lebesgue spaces (see [49]). In particular, from the above definition of the norm, we can directly get the following results: For any measurable function p : Ω ⟶ ½p − , p + ⊂ ð2,∞Þ, where p ± are constants, we define the second space and the variable-exponent Lebesgue space which is a Banach space with the following Luxemburg norm: We also assume that p satisfies the following Zhikov-Fan condition for the local uniform continuity: there exist a constant M > 0 such that for all points x,y in Ω with jx − yj < 1/2 , we have the inequality In addition, k:k q and k:k H 1 ðΩÞ denote the usual L q ðΩÞ norm and H 1 ðΩÞ norm.
In order to obtain the main results, we give the following lemma firstly.
then min u k k for any u ∈ L pð:Þ ðΩÞ (2) Assume that m, n, p : Ω ⟶ ð1,+∞Þ are measurable functions satisfying Then, for all functions u ∈ L pð:Þ ðΩÞ and v ∈ L nð:Þ ðΩÞ, we have uv ∈ L mð:Þ ðΩÞ with uv k k m : Then, the embedding H 1 0 ðΩÞ = W 1,2 0 ðΩÞ↪L pð:Þ ðΩÞ is continuous and compact, and there is a constant We assume that the relaxation function g and the potential α satisfy the following assumptions: Hypothesis g, α: g, α : ℝ + ⟶ ℝ + are nonincreasing differentiable functions such that Hypothesis ξ: there exist a positive differentiable functions ξ satisfying Hypothesis pð:Þ: the function pð:Þ satisfies Hypothesis μ 1 and μ 2 : the constants μ 1 and μ 2 satisfy Calculating ðd/dtÞαðtÞðg ∘ uÞðtÞ with respect to t, it 3 Journal of Function Spaces shows that where As in [38,43], we present a new time-dependent variable to deal with the time delay term: Consequently, we have Therefore, problem (1) can be transformed into By the standard methods as in Section 3 of [50], we can easily prove the well-posedness of problem (1) presented as follows.

Main Asymptotic Theorem
Next, we will give the proof of Theorem 4.
The functional E of problem (25) is as follows: where ξ and λ are positive constants and they satisfy The most important key to solve problem (1) is to obtain a result that concerns the asymptotic stability of solutions.
The main result is as follows.
Journal of Function Spaces Lemma 5. If u is a solution of problem (25). Then, Proof. Using the same idea as in [50], multiply the first equation in (25) By zð1, tÞ = u t ðt − τÞ and the Young inequality, we get From (23), we have Comparing (31) and (32), we obtain Setting by condition (28), we derived the desired inequality (30).
holds, EðtÞ may not be nonincreasing.

Lemma 7.
Assume that u be a solution of problem (25). Then, where l 0 and l as in (17).

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Now, we give a modified functional: where ε 1 , ε 2 , and N are positive constants. In fact, L is equivalent to E by the following lemma.

Lemma 8.
There exists C 1 , C 2 > 0 such that Proof. By the Poincaré theorem and Young inequality, we have the following results through integrating by parts: where c * as in Lemma 1, taking C 1 = N − Cðε 1 + ε 2 Þ and C 2 = N + Cðε 1 + ε 2 Þ, provided ε 1 and ε 2 are sufficiently small, and the proof is completed.
Proof. By the first equation of (25), we differentiate (42), and then we have By the Hölder inequality, Sobolev-Poincaré inequalities, and (17), we estimate the second part of the right-hand side in (47).

Data Availability
No data is used in the manuscript.

Conflicts of Interest
The authors declare that they have no conflicts of interest.