General decay for a system of viscoelastic wave equation with past history, distributed delay and Balakrishnan-Taylor damping terms

: The subject of this research is a coupled system of nonlinear viscoelastic wave equations with distributed delay components, inﬁnite memory and Balakrishnan-Taylor damping. Assume the kernels g i : R + → R + holds true the below

Several applications and real-world issues are frequently affected by the delay, which transforms numerous systems into interesting research topics.Numerous writers have recently studied the stability of the evolution systems with time delays, particularly the effect of distributed delay.See [24][25][26].
Based on everything said above, one specific problem may be solved by combining these damping terms (distributed delay terms, Balakrishnan-Taylor damping and infinite memory), especially when the past history and the distributed delay are added.We shall attempt to throw light on it since we think it represents a fresh topic that merits investigation and analysis in contrast to the ones mentioned before.Our study is structured into multiple sections: in the second section, we establish the assumptions, notions, and lemmas we require; in the final section, we substantiate our major finding.

Fundamental knowledge
In this section of the paper, we will introduce some basic results related to the theory for the analysis of our problem.Let us take the below: (G1) h i : R + → R + are a non-increasing C 1 functions fulfills the following and The functions G i (t) are strictly increasing and convex of class C 2 (R + ) on (0, ], r ≤ g i (0) or linear in a manner that in which ζ i (t) are a C 1 functions fulfilling the below (G4) f i : R 2 → R are C 1 functions with f i (0, 0) = 0, and one can find a function F in a way that Take the below and Lemma 2.1.(Sobolev-Poincare inequality [27]).Assume that 2 ≤ q < ∞ for n = 1, 2 and 2 ≤ q < 2n n−2 for n ≥ 3.Then, one can find c * = c(Ω, q) > 0 in a manner that Moreover, choose the below as in [26]: with rx t (z, ρ, r, t) + x ρ (z, ρ, r, t) = 0, sy t (z, ρ, r, t) + y ρ (z, ρ, r, t) = 0 x(z, 0, r, t) = v t (z, t), y(z, 0, r, t) = w t (z, t). (2.7) Take the auxiliary variable (see [28]) Then Rewrite the problem (1.1) as follows where (z, ρ, r, t) ∈ Ω × (0, 1) (2.10) In the upcoming Lemma, the energy functional will be introduced. (2.11) The above fulfills the below ≤ 0, (2.12) Proof.To prove the result, we take the inner product of (2.9) with v t , w t and after that integrating over Ω, the following is obtained Using mathematical skills, the following is obtained further simplification leads us to the following . (2.15) The following is obtained after calculation (2.16) In the same way, we have Now, multiplying the equation (2.9) by −x|β 2 (r)|, −y|β 4 (r)|, and integrating over Ω × (0, 1) × (τ 1 , τ 2 ) and utilizing (2.7), the below is obtained Similarly, we have Here, we utilize the inequalities of Young as and Finally, we have Thus, after replacement of (2.14)-(2.22)into (2.13),we determined (2.11) and (2.12).As a result, we obtained that E is a non-increasing function by (2.2)-(2.5),which is required.
Theorem 2.3.Take the function U = (v, v t , w, w t , x, y, η t , ϑ t ) T and assume that (2.1)-(2.5)holds true.Then, for any U 0 ∈ H, then one can find a unique solution U of problems (2.9) and (2.10) in a manner that

Analysis of stability
Here, the stability of the systems (2.9) and (2.10) will be established and investigated.For which the following lemma is needed Lemma 3.1.Let us suppose that (2.1) and (2.2) fulfills. where Proof.
which is obtained through Young's inequality (Eq 3.1).
It is mentioned in [12] that one can find a positive constant β, β in a manner that in which Proof.As the function E(t) is decreasing and utilizing (2.11), we have the following for any t, s ≥ 0. Further, we have in which β = max{ 4E(0) l 1 , 2} and µ(t) = ∞ 0 g 1 (t + s)(1 + Ω ∇u 2 0 (z, s)dz)ds.In the same way, we can deduce that in which β = max{ 4E(0) l 2 , 2} and In the upcoming part, we set the following and and (3.10) Lemma 3.4.In (3.8), the functional Ψ(t) fulfills the following Proof.To prove the result, differentiate (3.8) first and then apply (2.9), we have the following
Proof.To prove the result, simplification of (3.9) and (2.9) through mathematical skills leads us to the following Electronic Research Archive Volume 30, Issue 10, 3902-3929.Here, we will find our the approximation of the terms of the RHS of (3.18).Using the well-known Young's, Sobolev-Poincare and Hölder's inequalities on (2.1), (2.11) and Lemma 3.1, we proceed as follows and ) In the same, we obtained the following and to find the approximation of J 61 , we have 2) implies that In the same steps, the estimation of J i2 , i = 1, .., 6 are obtained and Proof.To prove the result, using Θ(t), and (2.9), we obtained the following Utilizing x(z, 0, r, t) = v t (z, t), y(z, 0, r, t) = w t (z, t), and e −r ≤ e −rρ ≤ 1, for any 0 < ρ < 1, moreover, select γ 1 = e −τ 2 , we have applying (2.4), the required proof is obtained.
In the next step, we below functional are introduced
Lemma 3.7.Let us suppose that (2.1) and (2.2) satisfied.Then, the functional F 1 = A 1 + A 2 and fulfills the following Proof.We can easily prove this lemma with the help of Lemma 3.7 in [13] and Lemma 3.4 in [15].Now, we have sufficient mathematical tools to prove the below mentioned Theorem.
We select the various constants at this point such that the values included in parenthesis are positive in this stage.Here, putting Thus, we arrive at In the upcoming, we select σ in a manner that σ < min l 0 4 , ll 0 8(ξ 0 + g 0 2 + c l) .
After that, we take N 2 in a way that and take N 3 large enough in a way that As a result, for positive constants d i , i = 1, 2, 3, 4, 5, (3.33) can be written as We know that κg 2 i (s) κg i (s)−g i (s) ≤ g i (s), then from from Lebesgue Dominated Convergence, we have the below which leads to lim κ→0 + κC κ = 0.As a result of this, one can find 0 < κ 0 < 1 in a manner that if κ < κ 0 , then By the fact e −ρr < 1 and (2.2), we have the below Here, set κ = 1 2N and take N large enough in a manner that for some k 2 > 0, and for some c 5 , c 6 > 0, we have After that, the below cases are considered: Case 3.9.G i , i = 1, 2 are linear.Multiplying (3.40) by ζ 0 (t) = min{ζ 1 (t), ζ 2 (t)}, we find

Conclusions
The purpose of this work was to study when the coupled system of nonlinear viscoelastic wave equations with distributed delay components, infinite memory and Balakrishnan-Taylor damping.Assume the kernels g i : R + → R + holds true the below g i (t) ≤ −ζ i (t)G i (g i (t)), ∀t ∈ R + , for i = 1, 2, in which ζ i and G i are functions.We prove the stability of the system under this highly generic assumptions on the behaviour of g i at infinity and by dropping the boundedness assumptions in the historical data.This type of problem is frequently found in some mathematical models in applied sciences.Especially in the theory of viscoelasticity.What interests us in this current work is the combination of these terms of damping, which dictates the emergence of these terms in the problem.In the next work, we will try to using the same method with same problem.But in added of other dampings.