Asymptotic stability for solutions of a coupled system of quasi-linear viscoelastic Kirchho ff plate equations

: In this manuscript, we study the asymptotic stability of solutions of two coupled quasi-linear viscoelastic Kirchho ff plate equations involving free boundary conditions, and accounting for rotational forces


Introduction
A coupled system of two Kirchhoff plate equations is considered: y(x, 0) = y 0 (x), y t (x, 0) = y 1 (x), z(x, 0) = z 0 (x), z t (x, 0) = z 1 (x) in Ω, where Ω is a bounded domain of R 2 with a smooth boundary Γ = ∂Ω = Γ 0 ∪ Γ 1 , such that Γ 0 ∩ Γ 1 = ∅, the initial data y 0 , y 1 , z 0 and z 1 lie in appropriate Hilbert space.The symbols y t and y tt refer, respectively, to first order and second order derivatives ( with respect to t) of y, while ∆ and ∆ 2 are the Laplacian and Bilaplacian operators.The functions h i and f i (for i = 1, 2) verify some assumptions that will be given in the next section.ρ is a positive constant, x = (x 1 , x 2 ) is the space variable, and the operators B 1 and B 2 are defined by , where the constant 0 < µ < 1  2 is the Poisson coefficient.Here, ∂ ν stands for normal derivative, ν = (ν 1 , ν 2 ) is the unit outer normal vector to Γ and τ = (−ν 2 , ν 1 ) is a unit tangent vector.
Model (1.1) describes the interaction of two viscoelastic Kirchhoff plates with rotational forces, which possess a rigid surface and whose interiors are somehow permissive to slight deformations, such that the material densities vary according to the velocity [1].Each one of these two plates is clamped along Γ 0 , and without bending and twisting moments on Γ 1 .The analysis of stability issues for plate models is more challenging due to free boundary conditions and the presence of rotational forces, etc. [2].Moreover, in our case the source term competes with the dissipation induced by the viscoelastic term only.Therefore, it will be interesting to study this interaction [3].
We start off by reviewing some works related to quasi-linear wave equation and plate equation.Cavalcanti et al. [1] considered the following equation and proved the global existence of weak solutions and a uniform decay rates of the energy in the presence of a strong damping, of the form −γ∆u t acting in the domain and assuming that the relaxation function decays exponentially.Messaoudi and Tatar [3] studied (1.2) but without a strong damping (γ = 0).They showed that the memory term is enough to stabilize the solution.The global existence and uniform decay for solutions of (1.2), provided that the initial data are in some stable set, are obtained in [4] with the presence of a source term and with γ = 0. Later, in [5], for γ = 0, the authors investigated the general decay result of the energy of (1.2) with nonlinear damping.In [6], the author investigated (1.2) with weakly nonlinear time-dependent dissipation and source terms, and he established an explicit and general energy decay rate results without imposing any restrictive growth assumption on the damping term at the origin.For other related results for quasi-linear wave equations, we refer the reader to [7][8][9][10].For quasi-linear plate equations, we mention the work of Al-Gharabli et al. [11] where the authors studied the well-posedness and asymptotic stability for a quasi-linear viscoelastic plate equation with a logarithmic nonlinearity.Recently, Al-Mahdi [12] studied the same problem as in Al-Gharabli et al. [11], but with infinite memory.With the imposition of a minimal condition on the relaxation function, he obtained an explicit and general decay rate result for the energy.Very recently, in [13], the authors considered a plate equation with infinite memory, nonlinear damping, and logarithmic source.They proved explicit and general decay rate of the solution.
The stability of coupled quasi-linear systems has been discussed by many authors.Liu [14] considered two coupled quasi-linear viscoelastic wave equations.He showed that the viscoelastic terms' dissipations guarantee that the solutions decay exponentially and polynomially.Later on, with more general relaxation functions and specific initial data, He [15] extended the result of Liu [14].Recently, Mustafa and Kafini [16] considered the same problem and improved earlier results for a wider class of relaxation functions.In [17], the authors studied the same problem, but with nonlinear damping, and showed a general decay rate estimates of energy of solutions.Very recently, Pis ¸kin and Ekinci [18] generalized and improved earlier results by considering a degenerate damping.Finally, let's mention the recent works of Fang et al. [19] and Zhu et al. [20] that relate to our problem.
As I know, there is no work regarding quasi-linear plate equations.This paper seems to be the first that deals with this problem.
The structure of this paper is shown as follows: In Section 2, we present some presumptions that are necessary for the proof of essential results.The third section provides the proof of well-posedness of our system.The general energy decay result is stated and established in Section 4. The fifth section provides two examples that illustrate explicit formulas for the energy decay rates.A concluding section is given at the end.

Preliminaries
This part is devoted to give some necessary materials and assumptions for the proof of our key results.We define Denoting dx = dx 1 dx 2 , we define the bilinear form b : Firstly, we must recall Green's formula (see [2]): and a weaker version of it (see Theorem 5.6 in [21]) in the following form: We need the following lemma.
Remark 2.1.1.The condition (A1) guarantees the hyperbolicity of the first two equations in the system (1.1).
2. By (2.6) and the mean value theorem, we have for some positive constant d 1 and ) The energy functional is defined by Here, K(t) = K y (t) + K z (t) and P(t) = P y (t) + P z (t) + Ω F(y, z) dx represent, respectively, the kinetic and the elastic potential energy of the model.
We have the following dissipation identity: Proposition 2.1.
Proof.Multiplying (1.1) 1 by y t and (1.1) 2 by z t , summing the resultant equations and integrating over Ω to get 3) in (2.11), we get the desired result.□ Throughout this paper, c denotes a generic positive constant, and not necessarily the same at different occurrences.

Global existence
We begin this part by defining a weak solution of the system (1.1).
Proof.With the help of the Faedo-Galerkin approach, the existence is demonstrated.In order to achieve this, let {w j } ∞ j=1 be a basis of V. Define E m = span{w 1 , w 2 , ..., w m }.On the finite dimensional subspaces E m , the initial data are projected as follows: Considering the following solution which satisfies the following approximate problem in E m : This leads to a system of ordinary differential equations (ODEs) for unknown functions p k and q k .Hence, from the standard theory of system of ODEs, a solution (y m , z m ) of (3.2) exists, for all m ≥ 1,
A priori estimate 1: Let w = y m t in (3.2) 1 and w = z m t in (3.2) 2 .Combining the resultant equations and integrating on Ω to obtain where Noting, by (3.1), that Then, by integrating (3.3) over (0, t), 0 < t < t m , we get a constant M 1 > 0 that doesn't depend on t and m, satisfying Hence, t m can be replaced by some T > 0, for all m ≥ 1.
A priori estimate 2: Let w = y m tt in (3.2) 1 and w = z m tt in (3.2) 2 , adding the resultant equations, integrating on Ω , and using Young's inequality to obtain for all η > 0 Using Hölder's inequality, Sobolev's embedding, (2.7) and (3.4), one has for some M 2 > 0, Similarly, we obtain that Integrating (3.8) on (0, T ), and using (3.4) gives us (3.9) Choosing η small enough, such that and so that Then, we have for some constant M 3 > 0.
From (3.4) and (3.10), we conclude that y m , z m are uniformly bounded in L ∞ (0, T ; V), and y m tt , z m tt are uniformly bounded in L 2 (0, T ; W).
Hence, we can extract subsequence of (y m ) and (z m ), still denoted by (y m ) and (z m ) respectively, such that and y m tt ⇀ y tt , z m tt ⇀ z tt weakly in L 2 (0, T ; W). (3.16) Analysis of the non-linear terms: 1. Term f i (y m , z m ): We have that (y m ) and (z m ) are bounded in L ∞ (0, T ; V).This shows, by the use of the embedding of , the boundedness of (y m ) and (z m ) in L 2 (Ω × (0, T )).
Likewise, (y m t ) and (z m t ) are bounded in L 2 (Ω × (0, T )).Hence, by the use of the Aubin-Lions Theorem, we get, up to a subsequence, that y m → y and z m → z strongly in L 2 (Ω × (0, T )).
Then, y m → y and z m → z a.e in Ω × (0, T ), and, therefore, from (A3), On the other hand, we have (y m ) and (z m ) that are bounded in L ∞ (0, T ; L 2 (Ω)), then, by using (2.7) and (3.4), we get that f i (y m , z m ) is bounded in L ∞ (0, T ; L 2 (Ω)).This fact and (3.17) leads to and similarly where C * is a positive constant satisfying ∥u∥ 2 ≤ C * ∥∇u∥ 2 , for all u ∈ W. Then Letting m → +∞, the aforementioned convergence results give that for all w ∈ V.
Since the terms in the right hand side of (3.
So, y m t (x, 0) and z m t (x, 0) make sense and Consequently, the proof of local existence of weak solutions is complete.Besides, it is easy to see that l 1 b(y, y) which gives the globalness and boundedness of the solution of problem (1.1).□

From (A2), one has lim
One can easily check, for i = 1, 2, that for some constants a i > 0 and b i > 0. This implies that Proof of Theorem (4.1):The proof is divided into three steps.
Step 1: In this step, we give estimates for the derivatives ( with respect to t) of the functionals φ(t) and ψ(t) defined below by: with ) Proof.We have φ . By using (1.1), we obtain Since then, by the use of Cauchy-Schwarz's inequality and Young's inequality, we derive Inserting (4.9) in (4.8), we get that Similarly, we infer that Summing the last two inequalities, we get the desired inequality (4.7).□ Lemma 4.2.If (A1)-(A3) hold.The functional defined in (4.6) verifies, for any 0 < δ < 1 and for all t ≥ t 1 , along the solution of (1.1),

10)
Here h 0 = min Proof.Differentiating ψ 1 (t) with respect to t and using (1.1) 1 , we get Now, we estimate the terms in the right-hand side of (4.11) as follows: • Estimation of the term  (4.12) • Estimation of the term We have By combining (4.12)-(4.16),using the fact that − t 0 h 1 (s)ds ≤ −h 0 for all t ≥ t 1 and choosing δ 1 small enough, we derive the estimate (4.10).□ Repeating the calculations above with ψ 2 (t) yields Combining (4.10) and (4.17), we obtain the following result.
Step 2: The aim of this step is to establish the inequality (4.26).
Let's define the functional where N > 0. For N sufficiently large, one has that F ∼ E, i. e.

.24)
Since ξ is non-increasing, then by using (4.24), the functional It is obvious that F ∼ E, and then we get the existence of some positive constant m 1 , such that By applying Gronwall's Lemma, there exists a constant m 2 > 0, such that By taking 0 < λ < 1 sufficiently small, we get, for all t > 0, J 1 (t) < 1 and J 2 (t) < 1.One can easily check that K i (t) ≤ −cE ′ (t), for i = 1, 2.
2. Let h i (t) = p i (1+t) q i , i = 1, 2, where q i > 0 and p i > 0 is chosen such that, (2.4) holds true.One has, for i = 1, 2, where ξ i (t) = q i p 1 q i i and G i (t) = t q i +1 q i .

Conclusions
This paper focuses on the existence and the asymptotic stability of solutions for a system of two coupled quasi-linear Kirchhoff plate equations in a bounded domain of R 2 , subject only to viscoelasticity dissipative terms and with the presence of rotational forces and source terms.Each one of these two equations describes the motion of a plate, which is clamped along one portion of its boundary and has free vibrations on the other portion of the boundary.This work is motivated by previous results concerning coupled quasi-linear wave equations [14][15][16] and single quasi-linear plate equation [12,13].
As future works, we can change the type of damping by considering, for example, weak damping (of the form y t ), Balakrishnan-Taylor damping (of the form (∇y, ∇y t )∆y) or strong damping (of the form ∆ 2 y t ).