On the System of Coupled Nondegenerate Kirchhoff Equations with Distributed Delay: Global Existence and Exponential Decay

Laboratory of Operator Theory and PDEs: Foundations and Applications, Department of Mathematics, Faculty of Exact Sciences, University of El Oued, El Oued, Algeria Department of Mathematics, College of Sciences and Arts, ArRas, Qassim University, Buraydah, Saudi Arabia Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella, Oran, Algeria Laboratory of Pure and Applied Mathematics, Amar Teledji Laghouat University, Algeria Department of Computer Sciences, College of Sciences and Arts, ArRas, Qassim University, Saudi Arabia Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt


Introduction
Let H = Ω × ðτ 1 , τ 2 Þ × ð0, ∞Þ, in this work, we consider where under the initial and boundary conditions where Ω be a bounded domain in ℝ n with smooth boundary ∂Ω, l > 0 and Δ is the Laplacian operator, and the functions μ 1 , μ 2 : ½τ 1 , τ 2 ⟶ ℝ are bounded, with 0 ≤ τ 1 < τ 2 , and the relaxation functions are denoted by g 1 , g 2 . The function M is given by with a, b > 0, and γ ≥ 1, and the functions f 1 , f 2 will be defined later.
In 1976, Kirchhoff developed an equation describing the vibrations produced by a fixed series at its end, since it is considered a generalization of the d'Alembert equation, and it belongs to the wave equation models. Over time, many researchers and authors addressed these issues and problems with their continuous and rapid development, for example, see [1][2][3][4].
We, under appropriate conditions, obtained the global existence of solutions, and we proved the exponential stability result of the system.
And we divided the paper into the following: in the second part, we set out the necessary hypotheses and the main result; in the third part, we prove the global existence of solutions, while in the fourth part, we present our result for exponential stability.

Preliminaries
In this section, we set the necessary hypotheses for proving the main result.

Journal of Function Spaces
Integrating (41) over ð0, tÞ, we get At this stage, choosing η > 0 such that we find We have from (17) and (49) that there exist subsequences ðu k Þ of ðu m Þ and ðv k Þ of ðv m Þ such that We work now with the nonlinear term. From (17), we find u kt j j l u kt where C 4 depends only on C * , C 1 , T, l.
And from the theorem of Aubin-Lions (see Lions [23]), we deduce that there exists a subsequence of ðu k Þ, given by ðu k Þ, such that we get Hence, Thus, using (46) and (48) and the Lions lemma, we derive Similarly, which implies The sequences ðu k Þ and ðv k Þ satisfy We have Noting that ðl/2pÞ + ð1/2qÞ + ð1/2Þ = 1, by applying the generalized Hölder's and Young's inequalities, and (8), we get

Exponential Decay
In this section, the stability result of the system (13)-(15) is proved.
We need the following lemmas.

Lemma 3. The functional
satisfies Proof.
(1) By applying the inequalities of Young and Poincare', we find (2) Direct computation using integration by parts, we get estimate (65) easily follows by using MðrÞ ≥ a, Young's inequality for ε 1 > 0, and (8).