Global Existence and Decay for a System of Two Singular Nonlinear Viscoelastic Equations with General Source and Localized Frictional Damping Terms

Department of Mathematics, College of Sciences and Arts, Al-Rass, Qassim University, Saudi Arabia Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella., Algeria Department of Mathematics, College of Exact Sciences, University of Tebessa, Tebessa 12002, Algeria Mascara University, Faculty of Economics, Mascara, Algeria Department of Mathematical Sciences, College of Applied Science, Umm Al-Qura University, Saudi Arabia


Introduction
The evolution problem with integral conditions is related with many branches of sciences ( [1][2][3][4][5][6]). Cause of this, interest in it occurs naturally in inflation cosmology, nuclear physics, supersymmetric field theories, and quantum mechanics (see for example [2,7]). Later, by the motivation of this work, some authors gave necessary and sufficient conditions for the hyperbolic equation with source term (see, e.g., [8][9][10]).
The problems related with localized frictional damping have extensively studied by many teams as [11], where the authors obtained an exponential rate of decay for the solution of the viscoelastic nonlinear wave equation: for damping term aðxÞu t may be null for some part of the domain.
We used the techniques in [11]; we have proven in [8] the existence of a global solution using the potential well theory for the following viscoelastic system with nonlocal boundary condition and localized frictional damping Very recently, in ( [9]), we study the following singular one-dimensional nonlinear equations that arise in generalized viscoelasticity with long-term memory:

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In view of the articles mentioned above in ( [8,9,11]), much less effort has been devoted to the existence of a global solution to the system of two singular nonlinear equations which arise in generalized viscoelasticity with localized frictional damping terms using the potential-well theory. Moreover, we prove a general decay result by constructing a Lyapunov functional and use it together with the perturbed energy method.
The structure of the work is as follows: To facilitate the description, firstly in Section 2, we give the fundamental definitions and theorems on function spaces that will be needed in the body of the paper and state the local existence theorem. In Section 3, the energy function EðtÞ is defined and proved to be a nonincreasing function of time. Finally, the main result is obtained, which gives the general decay conditions:

Preliminaries
Let L p x = L p x ðð0, LÞÞ be the weighted Banach space equipped with the norm when p = 2, we get a Hilbert space, and we denote by H = L 2 x , it provided with the finite norm , LÞÞ be the Hilbert space equipped with the norm We get the following lemma by combining the Poincare inequality and (see [8]).

Lemma 1.
Let V 0 space defined as follows Then, for 2 ≤ p 4, we have where C * is a constant depending on L and p only and for p = 2, C * = C p is the Poincare constant.
Remark 2. It is clear that kuk V0 = ku x k H defines an equivalent norm on V 0 . The next theorem confirms that our problem has a local solution under some condition on p and the relaxation func-tions g i , the proof can be established by following the argument of [12]. Theorem 3. We take ðu 0 , v 0 Þ ∈ V 2 0 and ðv 1 , v 2 Þ ∈ H 2 . If p < 3 and then, there exists t * >0 small enough such that the problem (1) has a unique local solution Remark 4. The condition on p is needed so that the embedding of V 0 in L 2 x is Lipchitz. We need the following assumptions to get our results. (G 1 ) For i = 1, 2, g i ðtÞ: ℝ + ⟶ ℝ + is a nonincreasing C 2 function such that and where ξðtÞ is a positive differentiable function. It satisfies for some positive constant l Furthermore, for any t 0 > 0 and 1 < σ < 3/2, there exists a positive constant C σ such that where a, b > 0 are constants and r > −1.
One can easily verify that Proof. From Minkowski inequality, we have We apply successively Holder's and Young's inequalities we obtain We combine the two previous inequalities and the embedding V ∩ V 0 ð0, LÞ, ↪L 2ðr+2Þ x ð0, LÞ, we get (20).

Lemma 6.
There exist two positive constants Λ 1 and Λ 2 such that Proof. It is clear that Applying Young's inequality with exponents q = ð2r + 3Þ/ ðr + 1Þ, q ′ = ð2r + 3Þ/ðr + 2Þ, in the last term in the above inequality, we get Consequently, by using Poincaré's inequality and (20), we obtain Similarly, we get the inequality for f 2 . The proof is completed.
We define the energy function as where Lemma 7. Let ðu, vÞ be the solution of system (1) then for all Hence, EðtÞ is a nonincreasing function.

Global Existence
In order to state and prove the global existence, we set the following notation we remark that Lemma 8. Assume that (G 1 ), (G 2 ), and (20) hold also for any where Then, there exists t * > 0 such that Proof. Since Ið0Þ > 0, then from the continuity of I(t), there exist t m ≤ t * such that IðtÞ ≥ 0 for all t ∈ ½0, t m Þ; this implies that we have a maximum time value noting T m such that From formulas of JðtÞ and IðtÞ together with (G 1 ), we have hence,

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Recalling Lemma 5 and (41), we get we deduce that IðtÞ > 0, ∀t ∈ ½0, T m Þ. By repeating the procedure, T m is extended to t * .
Proof. To achieve the proof of this theorem, it suffices to show that ku in the other hand and for the definition of IðtÞ, we have we introduce (49) into (50), we get by using (14), (15), and (41), (51) yields So where τ ≔ max 2, The proof is completed.

Decay of Solutions
Throughout this section, we will study the asymptotic behavior of solutions' decay by constructing a suitable Lyapunov function; to do so, for N, ε 1 , and ε 2 are positive constants, we define the following function as x Journal of Function Spaces with h ∈ C 1 ð½0, LÞ, hð0Þ = hðLÞ = 0, ðxhðxÞÞ ′ ≤ x, In the first step, we prove the equivalence between FðtÞ and EðtÞ given in the following lemma.
Lemma 10. For a choice of ε 1 and ε 2 small enough, we find two positive constants α 1 and α 2 such that Proof. By using Young inequality, follow by recalling Lemma 1 and the fact that 0 < ξðtÞ ≤ ξð0Þ, we get ξ t ð Þ 2 combining (60)−(66) in (55), we get If we choose ε 1 , ε 2 small enough, and N large enough we find α 1 , α 2 > 0 such that (59) holds true. Now, we state a Lemma corresponding to the boundness of ðgov x ÞðtÞ. It will be used in the calculus. Lemma 11. Let w ∈ L ∞ ðð0, TÞ ; HÞ be such that w x ∈ L ∞ ðð0, tÞ ; HÞ and g be a continuous function on ½0, T and suppose that. Then, there exists a constant C > 0 such that ∀0 < θ < 1 and ρ > 1: ð68Þ Proof.
In the next, we present three lemmas in which we give an upper bound of each derivative's functions in FðtÞ.
Proof. A derivation of (57) gives 9 Journal of Function Spaces by using Liebniz's formula, we get Recalling the differentials equation in (1), we get We will estimate all term in (86) by Young's inequality, Lemma 1, (G 1 ), and (G 2 ). and and Journal of Function Spaces where By the same technique, we obtain estimations on integrals corresponding to v, g 2 , and f 2 where A combination of (87)−(93) into (86) yields (83).
Next Theorem show that solutions decreases exponentially with respect to ξðtÞ and σ.