Abstract

This paper studies the system of coupled nondegenerate viscoelastic Kirchhoff equations with a distributed delay. By using the energy method and Faedo-Galerkin method, we prove the global existence of solutions. Furthermore, we prove the exponential stability result.

1. Introduction

Let , in this work, we consider where under the initial and boundary conditions where be a bounded domain in with smooth boundary , and is the Laplacian operator, and the functions are bounded, with , and the relaxation functions are denoted by . The function is given by with , and , and the functions 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 [1–4].

As for viscoelasticity, it is possible to delve into the following works for further clarification [3–10].

Also, the time or delay recorded in many natural and physical phenomena, especially problems resulting from vibrations, is an important factor for stability in general. And it has been studied extensively by many authors, including [5–7, 11–21]. Recently, in the presence of the varying delay, Mezouar and Boularrass studied system (1); for more information, see [22]. Based on these works, we in this work expand the results in [22] by adding the term of distributed delay.

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.

2. Preliminaries

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

We need the following assumptions:

(A1) are functions satisfying

(A2) satisfying

(A3) The number satisfying and

(A4) where such that and are onjugate and satisfy

We set the notations

As in [17], we introduce the new variables We have

Consequently, problem (1) is equivalent to where with the initial and boundary conditions

We need the following lemma.

Lemma 1. The energy functional , given by satisfies where ,
and .

Proof. Multiplying equation (13)1,2 by , and we use (15), one gets And multiplying equation (13)3 by , and integrating the result over , one gets Similarly, multiplying equation (13)4 by , we find by using the inequalities of Young and Cauchy-Schwartz for , we have Similarly, we get By summing (18)–(20) and using (21) and (22), and choosing such that , we find (16) and (17). This completes the proof.

3. Global Existence

Theorem 2. Suppose that (5)–(8) hold. Then, given , , and , there exists a weak solution of problem (13)–(15) such that where

Proof. Let the Galerkin basis , for , we set The sequences are defined for by Then, taking by over and denoting Given initial data , , and , we define the approximations It investigates the following problem: with initial conditions which satisfies Noting that , by using Hölder’s inequality, we get As (8) holds, using the embedding of Sobolev, the terms and in (29) make sense (see [22]).
First estimate.
As the sequences and converge and from (17) and Gronwall’s lemma, we get independent of such that where using (33) and (8), one gets The second estimate.
We multiply equation (29)1,2 by ; by summing from to , one gets By differentiating (29)3,4, we get And we multiply (37)1 by and (37)2 by ; by summing from to , we have Integrating the result (38) over with respect to , we obtain Summing (36) and (39) and using , we get At this point, we estimate the RHS of (40).
Integrating by parts, and using Young’s and Poincare’ inequalities, one gets Similarly, we get And, by using the inequality of Young, we get we have Similarly, we get substituting (41)–(45) into (40), and using (17), one gets where depends on .
Integrating (41) over , we get At this stage, choosing such that we find We have from (17) and (49) that there exist subsequences of and of such that We work now with the nonlinear term. From (17), we find where depends only on .
And from the theorem of Aubin-Lions (see Lions [23]), we deduce that there exists a subsequence of , given by , such that we get Hence, Thus, using (46) and (48) and the Lions lemma, we derive Similarly, which implies The sequences and satisfy We have Noting that , by applying the generalized Hölder’s and Young’s inequalities, and (8), we get As and are Cauchy sequences in (prove it as in [1]), then we get (59)1. Similarly, we get the convergence (59)2.
Multiplying (29) by and integrating the result over , we get We obtain (62) by the convergence of (50), (54), (56), and (59). This completes the proof.

4. 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 , Young’s inequality for , and (8).

Lemma 4. The functional satisfies and for any ,

Proof. (1)By using Young’s inequality and the conjugate exponents , , and Hölder’s inequality, we obtain Similarly, we get By combining (71)–(74), we find (69). (2)By derivation of , and integrating by parts and (15), we find Using Young’s, Cauchy-Schwarz, Hölder’s, and Poincaré’s inequalities, and , we obtain (70).

At this point, let us introduce the functional given by

Lemma 5. The functional satisfies where .

Proof. By derivation of , and using equations (13)3 and (13)4, we get Applying the equality , , and , for any , we get As is an increasing function, we have , for any .

Then, setting , we find (78).

Theorem 6. Assume (5)–(8) hold, then such that the energy functional (16) satisfies

Proof. We define the functional of Lyapunov where .
First, if we let then, by (64), (69), and (77), we get Consequently, which yields By derivation (82) and applying (17), (65), (70), (78), and (6), one gets where , , , ,, and .
At this stage, choosing two fixed numbers , , such that , and we choose small enough such that After that, we choose small enough such that Thus, we get using (16), estimates (91) and (86), respectively, we get for some
By the combination of (93) with (92), we obtain where Integrating the result (94) over , we find It follows from (95) that (81) holds. This completes the proof.

Data Availability

No data were used to support the study.

Conflicts of Interest

This work does not have any conflicts of interest.

Acknowledgments

The sixth author extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through General Research Project under Grant No. G.R.P-2/42. In addition, the third author would like to thank all the professors of the mathematics department at the University of Annaba in Algeria, especially his Professors/Scientists Pr. Mohamed Haiour, Pr. Ahmed-Salah Chibi, and Pr. Azzedine Benchettah for the important content of masters and Ph.D. courses in pure and applied mathematics that he received during his studies. Moreover, he thanks them for the additional help they provided to him during office hours in their office about the few concepts/difficulties he had encountered, and he appreciates their talent and dedication for their postgraduate students currently and previously.