On a Couple of Nonlocal Singular Viscoelastic Equations with Damping and General Source Terms: Blow-Up of Solutions

Under some given conditions, we prove the explosion result of the solution of the system of nonlocal singular viscoelastic with damping and source terms on general case. This current study is a general case of the previous work of Boulaaras.


Introduction
During the last decades, many nonlocal problems of deterministic and parabolic partial differential equations have been studied. These equations and their systems represent the modeling of many physical phenomena related to time. These constraints can be data measured directly at the boundary or give integral boundary conditions (for instance, see ).
In this work, we investigate the blow-up of the following system of nonlinear damping term: with a, b ∈ R, r≥−1 (we get a = b = 1), Q = ð0, αÞ × ð0, TÞ, α < ∞, T < ∞, and g 1 : ð Þ,g 2 : are given functions which will be specified later. The motivation of our work is because of some results regarding the following research paper: in [12], under some conditions suitable for the relaxation function, the author explained that solutions with initial negative energy explode in a finite time if p > m and continue to find if m ≥ p, for the following studied problem: movement of a flexible two-dimensional viscous body on the unit disk (i.e., radial solutions) and by using some density arguments and some prior estimates, the authors demonstrated the existence and uniqueness of a generalized solution to the following problem: where and f is the right-hand side that satisfied the Lipschitzian condition. Recently, in [3], the authors demonstrated the decay result of energy for a small enough initial data together with the explosion result of large initial data of the following singular problem: That is, they obtained the blow-up properties of local solution by Georgiev-Todorova method with nonpositive initial energy. More work followed up on similar nonlocal singular viscoelastic equations and systems in [8,9].
In this work, we continue the study on system (1). According to some given conditions, we prove the explosion result of the solution of the system of nonlocal singular viscoelastic with damping and source terms on general case, where we begin by giving basic definitions and theories about the function spaces we need, and then, we mention the theorem of local existence. Finally, we announce and prove the main result of our studied problem in (1).

Preliminaries.
In this section, we introduce some functional spaces and give some lemma's need for the remaining of this paper. Let L p x = L p x ðð0, αÞÞ be the weighed Banach space equipped with the norm Lemma 1 (Poincare-type inequality). For any u ∈ V 0 , where C P is some positive constant.
Remark 2. It is clear that kuk V 0 = ku x k H defines an equivalent norm on V 0 : where C * is a constant depending on α and p only. For the g 1 and g 2 functions, assumptions are as follows:(G1): g 1 ð:Þ, g 2 ð :Þ: R + ⟶ R + are two differentiable and nonincreasing functions with (G2): For all t ≥ 0, for T > 0 small enough. is a solution of problem((1)); then, the energy functional where Remark 6. Multiplying the first equation in((1))by xu t and the second equation in((1))by xv t integrating over ð0, αÞ , we obtain the following equation: The definition of the norm is as follows: From here, Thus, Lemma 7. There exist c 0 and c 1 positive constants such that

Blow-Up of Solution
In this section, we shall deal with the blow-up behavior of solutions for problem (1). We derive the blow-up properties of solutions of problem (1) with nonpositive initial energy by the method given in [1].
Proof. Since ðd/dtÞ½EðtÞ = E′ðtÞ ≤ 0, We define HðtÞ = −EðtÞ; then, ☐ We obviously substitute EðtÞ in (26); then, From (22) and (27), Thus, Equation (29) will then be used as an important data for proof of the theorem. Now, we define 3 Journal of Function Spaces for ε small enough and By differentiating (30), using (1) and By using Young inequality and from HðtÞ = −EðtÞ, we obtain where From (34), and for a 5 < min fα 1 , α 2 , α 3 , α 4 , 2ðr + 2Þg, To estimate the last term in (36), we apply the threeparameter Young inequality: a, b ≥ 0,ð1/rÞ + ð1/qÞ = 1 ,ab ≤ ðδ r /rÞa r + ðδ −q b q /qÞ,∀δ > 0. We take in this case: Journal of Function Spaces Substituting (38), (40), and (41) into (36), by organizing, we obtain Since integration in estimate (40) and (41) is performed over the space, the parameter δ 1 and δ 2 can be a function of time; we get them as follows: where k 1 > 0 and k 2 > 0 are sufficiently large constants to be specified further. By using (43) and (44) in (42), we have To estimate the last two terms in (45), we use (29); then, On the other hand, since r > max fm, mg from L 2ðr+2Þ x°L m+1 Substituting (47) into (46), By using we can estimate the following: Consequently, we have Similarly By using (51) and (52), forc · C = C′; we have From here, Thus, by applying (23), we obtain Substituting these inequalities in (54) and (55), in this case, With the combination of (59) and (60), we obtain Finally, and by considering (61), thus by organizing (45), we have which introduce the constant Taking sufficiently large k 1 > 0 and k 2 > 0 for the positive constant γ, we simplify (63) For fixed k 1 > 0, k 2 > 0, and γ > 0, we choose ε > 0 so small that the following inequality holds: Moreover, we assume that the initial data satisfy the estimate Journal of Function Spaces Then, from (65), we obtain the following inequality: On the other hand, in Equation (30), we take the1/ð1 − σÞ · powerof each side Twice by applying the following inequality to (69) we have where C > 0. Now, to estimate the last two terms in (71), we, respectively, apply Holder inequality, L 2ðr+2Þ x°L H x , and Young inequality; thus, Similarly, where ð1/θÞ + ð1/μÞ = 1. In these inequalities by collecting side by side, we obtain We choose μ = 2ð1 − σÞ, to get then ð α By applying (23), we can write From here, we obtain Thus, by considering (78) and the following in (71), we obtain Finally, by combining (68) and (80), we obtain the following ordinary differential inequality: obviously, where λ > 0 is a constant depending only C, ε, and 7 Journal of Function Spaces γ. This differential inequality integration over ð0, tÞ gives where we choose Hence,

Conclusions
The purpose of this paper is to study the explosion result of the solution of the system of nonlocal singular viscoelastic with damping and source terms on general case. This current study is a general case of the previous work of Boulaaras in ( [5]). In the next work, we will try to obtain the same result for the two-dimensional problem that allows a reasonable description of the phenomenon occurring in a threedimensional domain. Then, we will try to prove uniqueness results of the weak solution.

Data Availability
No data were used to support the study.

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