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Research Article | Open Access
Abdelbaki Choucha, Salah Mahmoud Boulaaras, Djamel Ouchenane, Salem Alkhalaf, Ibrahim Mekawy, Mohamed Abdalla, "On the System of Coupled Nondegenerate Kirchhoff Equations with Distributed Delay: Global Existence and Exponential Decay", Journal of Function Spaces, vol. 2021, Article ID 5577277, 13 pages, 2021. https://doi.org/10.1155/2021/5577277
On the System of Coupled Nondegenerate Kirchhoff Equations with Distributed Delay: Global Existence and Exponential Decay
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.
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].
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 . Based on these works, we in this work expand the results in  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.
In this section, we set the necessary hypotheses for proving the main result.
We need the following assumptions:
(A1) are functions satisfying
(A3) The number satisfying and
(A4) where such that and are onjugate and satisfy
We set the notations
As in , 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
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
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
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 ).
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