New general decay result for a system of two singular nonlocal viscoelastic equations with general source terms and a wide class of relaxation functions

We establish a new general decay result that improves most of the existing results in the literature related to this system. Our result allows for a wider class of relaxation functions, from which we can recover the exponential and polynomial rates when ki(s) = sp and p covers the full admissible range [1, 2).


Introduction
In this paper, we consider the following system: © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
where = (0, L), k i : [0, +∞) − → (0, +∞), (i = 1, 2), are non-increasing differentiable functions satisfying more general conditions to be mentioned later and ⎧ ⎨ ⎩ f 1 (u, v) = a|u + v| 2(r+1) (u + v) + b|u| r u|v| r+2 , where r > -1 and a, b > 0. Mixed nonlocal problems for parabolic and hyperbolic partial differential equations have received a great attention during the last few decades. These problems are especially inspired by modern physics and technology. They aim to describe many physical and biological phenomena. For instance, physical phenomena are modeled by initial boundary value problems with nonlocal constraints such as integral boundary conditions, when the data cannot be measured directly on the boundary, but the average value of the solution on the domain is known. Initial boundary value problems for second-order evolution partial differential equations and systems having nonlocal boundary conditions can be encountered in many scientific domains and many engineering models and are widely applied in heat transmission theory, underground water flow, medical science, biological processes, thermoelasticity, chemical reaction diffusion, plasma physics, chemical engineering, heat conduction processes, population dynamics, and control theory. See in this regard the work by Cannon [1], Shi [2], Capasso and Kunisch [3], Cahlon and Shi [4], Ionkin and Moiseev [5], Shi and Shilor [6], Choi and Chan [7], and Ewing and Lin [8]. In early work, most of the research on nonlocal mixed problems was devoted to the classical solutions. Later, mixed problems with integral conditions for both parabolic and hyperbolic equations were studied by Pulkina [9,10], Yurchuk [11], Kartynnik [12], Mesloub and Bouziani [13], Mesloub and Messaoudi [14,15], Mesloub [16], and Kamynin [17]. For instance, Said Mesloub and Fatiha Mesloub [18] obtained existence and uniqueness of the solution to the following problem: and proved that the solution blows up for large initial data and decays for sufficiently small initial data. Mesloub and Messaoud [14] considered the following nonlocal singular problem: and proved blow-up result for large initial data and decay results of sufficiently small initial data enough for p > 2. In [19], Draifia et al. proved a general decay result for the following singular one-dimensional viscoelastic system: where Q = (0, α) × (0, t) and p, q > 1. Piskin and Ekinci [20] studied problem (1) when the Bessel operator has been replaced by a Kirchhoff operator with a degenerate damping terms. They proved the global existence and established a decay rate of solution and also a finite time blow up. Recently, Boulaaras et al. [21] treated problem (1) and proved the existence of a global solution to the problem using the potential-well theory. Moreover, they established a general decay result in which the relaxation functions k 1 and k 2 satisfy Motivated by the above work, we prove a general stability result of system (1) replacing the condition (6) used in [21] by a more general assumption of the form: Our decay result improves all the existing results in the literature related to this system. This paper is divided into four sections. In Sect. 2, we state some assumptions needed in our work. Some technical lemmas will be given in Sect. 3. The statement with proof of the main result and some examples will be given in Sect. 4.

Preliminaries
In this section, we present some materials needed in the proof of our results. We also state, without proof, the global existence result for problem (1). Let Remark 2.1 Notice that u V 0 = u x L 2 x defines an equivalent norm on V 0 .

Local and global existence
In this subsection, we state, without proof, the local and global existence results for system (1), which can be proved similarly to the ones in [14,18] and [21].
Then problem (1) has a unique local solution.
For the global existence, we introduce the following functionals: and We notice that there exists t * > 0 such that Remark 2. 5 We can easily deduce from Lemma 2.3 that 2 and satisfies (16), then the solution of (1) is global and bounded.

Technical lemmas
In this section, we establish several lemmas needed for the proof of our main result.
Proof We prove inequality (19) for f 1 and the same result holds for f 2 . It is clear that From (20) and Young's inequality, with Consequently, by using (7), (12), (13) and the embedding V 0 → L 2(2r+3) , we obtain This completes the proof of Lemma 3.1.
Proof We will prove inequality (35) and the same proof also holds for (36). By Young's inequality and the fact that K 1 (t) = -k 1 (t), we see that Using the facts that K 1 (0) = 1 -1 and and the estimate where t 0 is introduced in Lemma 3.2 and = min{ 1 , 2 }.

General decay result
In this section, we state and prove our main result.
Proof Using Lemmas 3.6 and 3.7, we easily see that is nonnegative and, for any t ≥ t 0 , and, for some C > 0, Therefore, we arrive at ∞ 0 E(s) ds < +∞.
Without loss of the generality, we assume that I i (t) > 0, for any t > t 0 ; otherwise, we get an exponential decay from (38). We also define the following functionals: x ds and observe that Using (2.4), Assumption (A2), inequality (43) and Jensen's inequality, we obtain where¯ 1 is defined in Remark (2.3). Then, we have Similarly, we can have Thus, the estimate (40) becomes Set H = min{¯ 1 ,¯ 2 } and define the functional for ε 0 ∈ (0, ε) and t ≥ t 0 .
Using the fact that¯ i > 0,¯ i > 0 and E ≤ 0, we also deduce that F 1 ∼ E. Further, we get , for a.e t ≥ t 0 .
Recalling that E ≤ 0, then we drop the first and last terms of the above identity. Therefore, by using the estimate (45), we have , for a.e t ≥ t 0 .
In the sense of Young [23], we let¯ * i be the convex conjugate of¯ i such that * and it satisfies the following generalized Young inequality: By letting A = H(ε 0 ξ i (t) ), for i = 1, 2, and combining (46)-(48), we have, for almost every t ≥ t 0 , Multiplying the above estimate by ξ (t) = min{ξ 1 (t), ξ 2 (t)} > 0 and using the fact in (44), we get -cE (t), for a.e t ≥ t 0 .