General decay for weak viscoelastic equation of Kirchhoff type containing Balakrishnan–Taylor damping with nonlinear delay and acoustic boundary conditions

In this paper, we consider the general decay of solutions for the weak viscoelastic equation of Kirchhoff type containing Balakrishnan–Taylor damping with nonlinear delay and acoustic boundary conditions. By using suitable energy and Lyapunov functionals, we prove the general decay for the energy, which depends on the behavior of both σ and k.


Introduction
The objective of this work is to study the general decay of solutions for the weak viscoelastic equation of Kirchhoff type containing Balakrishnan-Taylor damping with nonlinear delay and acoustic boundary conditions |w t | ρ w tta 0 + b 0 ∇w 2 + b 1 (∇w, ∇w t ) ww tt + σ (t) t 0 k(ts) w(s) ds = |w| p-2 w in × R + , (1.1)
The equation (1.1) with b 0 = b 1 = 0 and a 0 = σ (t) = 1, = |w| p-2 w in × R + , w = 0 on (1.8) has been studied by Messaoudi and Tatar [16]. The case of ρ = 1 and b 1 = σ (t) = 0 in the absence of the dispersion term, the equation (1.1) reduces to the well-known Kirchhoff equation that has been introduced in [8] in order to describe the nonlinear vibrations of an elastic string. The model with Balakrishnan-Taylor damping (b 1 > 0) and k = 0, was initially proposed by Balakrishnan and Taylor in [2]. Several authors have studied the asymptotic behavior of the solution for the nonlinear viscoelastic Kirchhoff equations with Balakrishnan-Taylor damping (see [17,22,24] and references and therein). Recently, Al-Gharabli et al. [1] considered the following Balakrishnan-Taylor viscoelastic equation with a logarithmic source term They proved the general decay rates, using the multiplier method and some properties of the convex functions. Lian and Xu [11] investigated the problem (1.9) with weak and strong damping terms and ρ = b 0 = b 1 = k = 0. For σ (t) > 0, Messaoudi [15] studied the following viscoelastic wave equation The author obtained the general decay result that depends on the behavior of both σ and k. For other related works, we refer the readers to [3,13,14]. Since most phenomena naturally depend not only on the present state but also on some past occurrences, in recent years, there has been published much work concerning the wave equation with delay effects that often appear in many practical problems [18][19][20][21]. Feng and Li [7] proved the general energy decay for a viscoelastic Kirchhoff plate equation with a time delay. Lee et al. [9] showed the general energy decay of solutions for system (1.1)-(1.7) with σ (t) = 1 and q = 1.
Motivated by previous work, we study the general energy decay of solutions for the system (1.1)-(1.7) that depends on the behavior of the potential σ and the relaxation function k satisfying the suitable conditions. The acoustic boundary condition (1.4) and the coupled impenetrability boundary condition (1.3) were proposed by Beale and Rosencrans [5]. For physical application of acoustic boundary conditions, we refer to [4,6]. The stability of various models with acoustic boundary conditions has been discussed by many researchers [10,12,14,23]. The outline of this paper is as follows. In Sect. 2, we present some preparations and hypotheses for our main result. In Sect. 3, we obtain the general energy decay of the system (1.1)-(1.7) by using the energy-perturbation method.

Preliminary
In this section, we present some material that we shall use in order to prove our result. We denote by The Poincaré inequality holds in V , i.e., there exists a constant C * such that and there exists a constantC * such that For our study of problem (1.1)-(1.7), we will need the following assumptions.
(H1) The constants ρ and q satisfy and p satisfies For the relaxation function k and potential σ , as in [15], we assume that (H2) k, σ : R + → R + are nonincreasing differentiable functions such that k is a C 2 function and σ is a C 1 function satisfying where l and t 0 are suitable positive constants. There exists a nonincreasing differentiable function ζ : (H3) There exist three positive constants m 1 , g 1 , and h 1 such that We assume that the constants μ 1 and μ 2 satisfy μ 2 < μ 1 .
Examples of functions k and σ satisfying (H2) are for a, b > 0, to be chosen properly.
As in [19], let us introduce the function (2.10) By combining with the argument of [5], we now state the local existence result of problem (2.10), which can be obtained.

Main result
In this section, we state and show our main result. For this purpose, we define and ds. Now, we denote the modified energy functional E(t) associated with problem (2.10) by where ξ is a positive constant such that Note that this choice of ξ is possible from assumption (H4).

7)
where C 1 and C 2 are some positive constants.
Now, we will establish the general decay property of the solution for problem (2.10) in the case μ 2 < μ 1 . For this purpose, we define the functional (t) = ME(t) + εσ (t) 1 (t) + σ (t) 2 (t), (3.15) where M and ε are positive constants that will be specified later and (3.16) and 2 (t) = - Before we show our main result, we need the following lemmas.