Blow-up of solutions for a system of nonlocal singular viscoelastic equations with sources and distributed delay terms

In this paper, we investigate a scenario concerning a coupled nonlocal singular viscoelastic equation with sources and distributed delay terms. By establishing suitable conditions, we have proved that a ﬁnite-time blow-up occurs in the solution.

These issues arise in one-dimensional or longitudinal elasticity when considering longterm memory viscosity.The second integral represents the distributed delay terms, where τ 1 , τ 2 > 0 denote time delays, μ 2 , μ 4 are L ∞ functions.
Our work is motivated by the findings presented in the following papers: In [13], the authors examined a model depicting the motion of a viscoelastic twodimensional body on the unit disc, focusing on radial solutions.Furthermore, they established the uniqueness and existence of the generalized solution for the below stated nonlocal problem where Q = (0, 1) × (0, T) with right hand side f is a Lipshitzian function.
The findings presented in [14] indicate the occurrence of blow-up for large initial data and demonstrate decay outcomes for sufficiently small initial data, which are applicable to the subsequent nonlocal singular problem They acquired the blow-up properties of local solutions utilizing Georgiev-Todorova method, even with negative initial energy.Authors in [11], employed the direct method to prove blow-up of solutions under appropriate conditions on initial data [23].The authors in [26], extended the previous result to systems with higher dimensions and get additional blow-up results.The authors in [15], substituted the source terms f 1 (u, v) and f 2 (u, v) in the studied system in [26], respectively by |v| q+1 |u| p-1 u, |u| p+1 |v| q-1 v and the Bessel operator and with two different functions g(.).Moreover, it is augmented by both nonlocal and classical condition.Furthermore, in [17], the authors investigated the identical problem presented in [15], where they derived a nonlinear source of polynomial nature.This source is capable of inducing solutions to blow up within a finite time frame, even in the existence of enhanced damping u t .They considered three different cases regarding the sign of the initial energy.Pişkin and Ekinci [22] have addressed problem (1) by substituting Bessel operator with the Kirchhoff operator featuring degenerate damping terms.They employed a technique identical to that used to establish the global existence and provide a decay rate for solutions, as well as demonstrate a finite-time blow-up when the behavior of decreasing relaxation functions is stated as: and ξ (t) satisfies ∞ 0 ξ (s) ds = +∞, ∀t > 0.
Boulaaras et al. [18] investigated the subsequent system, which consists of two singular one-dimensional nonlinear equations arising in generalized viscoelasticity, featuring longterm memory, nonlocal boundary conditions, and general source terms By employing potential-well theory the authors proved the existence of a global solution for the problem.And in the same vein, in [5] Boularaas and Mezouar proved the existence and decay of solutions of a singular nonlocal viscoelastic system featuring nonlocal boundary conditions, localized damping term and linear source term.In domain of blowup phenomena, the authors in [27] investigated the finite-time blow-up of solutions for an initial boundary value problem with nonlocal boundary conditions, pertaining to a system of nonlinear singular viscoelastic equations.Other works in the same vein can be found here [1-4, 7, 10-12, 24, 25].The influence of delay frequently emerges in numerous applications and practical issues, transforming various systems into distinct problems warranting investigation.Recently, numerous authors have examined asymptotic behavior, stability, and blow-up phenomena of solutions in evolution systems with time delay.Refer to works by [6,8,9,19] for further details.
Motivated by the aforementioned works, in this study, we expand upon the earlier investigation outlined in [15] to encompass singular one-dimensional nonlinear viscoelastic equations with source and distributed delay terms.Specifically, we delve into the blow-up phenomenon of solutions with negative initial energy for problem (1).
In the following, let c, c i , C > 0, are positive constants.Our paper is structured as follows: In the subsequent section, we establish concepts, lemmas, and hypotheses essential for our analysis.In Sect.3, we state and prove the blowup phenomenon of solutions.

Preliminaries
In this section, we present the following definitions, symbols, spaces and lemmas that we utilize throughout the paper.
Let L p x = L p x (0, L) represent the weighted Banach space having norm .
Hilbert space of square integral functions is denoted by H = L 2 x (0, L) having the finite norm .
Hilbert space is represented by and Theorem 1 (See [1]) For any v in V 0 and 2 < p < 4, we have where C * is a constant depending on L and p only.
We demonstrate the blow-up outcome given the following appropriate assumptions.(A1) g 1 , g 2 : R + → R + are decreasing and differentiable functions such that (A2) There exists a constants ξ 1 , ξ 2 > 0 such that By combining arguments of other studies [16,20,22] with used the Faedo-Galerkin method, we get the local existence theorem.
Theorem 2 Let (3), (4), and (5) hold.Assume that Then, for any Lemma 2 There exists a function F(u, v) such that where We take a 1 = b 1 = 1 for convenience.
Lemma 3 [21] There exist two positive constants c 0 and c 1 such that Taking new variable as in [19], Consequently, the problem (1) is equivalent to We define the energy functional.

Blow-up
In this segment, we establish the blow-up outcome for the solution of problem (1).
Then the proof is complete.