Global Well-Posedness and Stability for a Viscoelastic Plate Equation with a Time Delay

A plate equation with a memory term and a time delay term in the internal feedback is investigated. Under suitable assumptions, we establish the global well-posedness of the initial and boundary value problem by using the Faedo-Galerkin approximations and some energy estimates. Moreover, by using energy perturbation method, we prove a general decay result of the energy provided that the weight of the delay is less than the weight of the damping.

Equation (1) with the memory term ∫  0 ( − )Δ(), where the function  is called kernel, can be regarded as a fourth order viscoelastic plate equation with a lower order perturbation, and it can be also regarded as an elastoplastic flow equation with some kind of memory effect.
In this paper, we consider the following initial conditions: (, 0) =  0 () ,   (, 0) =  1 () ,  ∈ Ω,   (,  − ) = ℎ 0 (,  − ) ,  ∈ Ω,  ∈ (0, ) and the following boundary conditions: Fourth order equations with lower order perturbation are related to models of elastoplastic microstructure flows.For the single plate equation without delay, that is,  2 = 0, as considered by Woinowsky-Krieger [1], the author first proposed the one-dimensional nonlinear equation of vibration of beams, which is given by where  is the length of the beam and , ,  are positive physical constants.The nonlinear part of (4) represents for the extensible effect for the beam whose ends are restrained to remain in a fixed distance apart in its transverse vibrations.
In recent years, many mathematical workers studied some systems with time delay effects.Datko et al. [21] studied the following system: By using an observability inequality, they proved the exponential stability for the energy when  2 <  1 .Subsequently, Xu et al. [22] obtained the same result as in [21] for the one space dimension by using the spectral analysis approach.Later on, Kirane and Said-Houari [23] considered a viscoelastic wave equation with a delay term in internal feedback with initial conditions and boundary value conditions of Dirichlet type.Under suitable assumptions on the relaxation function and some restriction on the parameters  1 and  2 , they established the global well-posedness of the system.Moreover, under the assumption  2 ≤  1 between the weight of the delay term in the feedback and the weight of the term without delay, the authors proved a general decay of the total energy of the system.For more some results concerning the different boundary conditions under an appropriate assumption between  1 and  2 , one can refer to Nicaise and Pignotti [24], Nicaise et al. [25], Nicaise and Valein [26], and the references therein.Equation ( 1) is a plate equation with a memory term and a time delay term in the internal feedback.Noting that  1 ̸ = 0, we know that it is a plate equation with weak damping.For viscoelastic plate equations, it is well known that one considered a memory of the form ∫  0 ( − )Δ 2 () (see, e.g., [10,27,28]).However, because the main dissipation of the system (1)-( 3) is given by a weak damping   , here we consider a weaker memory, acting only on Δ.To the best of our knowledge, the global well-posedness and energy decay for system (1)-(3) were not previously considered.So the objective of this work is to establish the global well-posedness and stability of initial boundary value problem (1)- (3).The main dissipation of the system (1)-( 3) is given by a weak damping   , which makes the analysis in this work different from [16], because the authors considered the case of a strong damping −Δ  in [16].
The outline of this paper is as follows.In Section 2, we give some preparations for our consideration and our main results.In Section 3, we establish the global posedness of the system by using the Faedo-Galerkin approximations and some energy estimates.In Section 4, we will show a general decay result of the energy by using energy perturbation method provided that the weight of the delay is less than the weight of the damping.

Preliminaries and Main Results
In this section, we give some preparations for our consideration and our main results.

The Global Well-Posedness
In this section, we will prove the global existence and the uniqueness of the solution of problem ( 17)-( 18) by using the classical Faedo-Galerkin approximations along with some priori estimates.We only prove the existence of solution in (i).For the existence of stronger solution in (ii), we can use the same method as in (i) and one can refer to Andrade e al. [16] and Jorge Silva and Ma [28].

A Priori
Noting the following fact: where we know that ( Multiplying the second approximate equation of ( 28) by (/)ℎ   and then integrating over (0, ) × (0, 1), we obtain A straightforward calculation gives Then we have the following cases.
We can prove the continuous dependence and uniqueness for weak solutions by using density arguments (see, e.g., Cavalcanti et al. [27]) which also can be found in Lions [29] (Chapter 1, Theorem 1.2) by using a regularization method and in Pata and Zucchi [31] or Giorgi et al. [32] by using the mollifiers.
This ends the proof of Theorem 1.

General Decay
In this section, we will establish the decay property of the solution for problem ( 17)- (18) in the case  2 <  1 .Motivated by [27,33], we use a perturbed energy method and suitable Lyapunov functionals.We first consider stronger solutions.Define the modified energy by where  is a positive constant satisfying (19).It follows from ( 9) and ( 14) that that is, (66) Proof.For the same argument as (41) in Section 3.2, we can easily get (66).Here we omit the detailed proof.Now we define the following functional: Then we have the following lemma.
Lemma 4.Under the assumptions in Theorem 2, the functional Φ() defined in (67) satisfies that, for any  > 0, where  1 is the Poincaré constant.
In order to handle the term (, , ), we introduce the functional Then we have the following estimate.
Lemma 5.Under the assumptions in Theorem 2, the functional Ψ() defined in (73) satisfies that where  1 is a positive constant.
Proof.Differentiating (73) with respect to  and using the second equation ( 17 ( Then it is easy to verify that there exists a constant  1 > 0 satisfying (74).