Research Article | Open Access
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.
In this paper, we are concerned with the following plate equation with a memory term and a time delay term in the internal feedback:where is a bounded domain with smooth boundary . Here is a function satisfying suitable conditions (see below), , are positive constants, and represents the time delay.
Equation (1) with the memory term , 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: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, , as considered by Woinowsky-Krieger , 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. A more general equation of (4) readswhere is a function satisfying some conditions. There are so many existing results concerning global existence, stability, and long-time dynamics for (5); we would like to refer the reader to de Brito , Cavalcanti et al. [3, 4], Ma , Ma and Narciso , de Lacerda Oliveira and de Lima , J. Y. Park and S. H. Park , Patcheu , Rivera [10, 11], Tusnal , Vasconcellos and Teixeira , Yang [14, 15], and the references therein. Very recently, Andrade et al.  investigated a viscoelastic plate equation with -Laplacian and memory terms with strong dampingwhere is the -Laplacian operator. Under suitable assumptions on the memory kernel and a forcing term , the authors proved the existence of weak solutions by using Faedo-Galerkin approximations, the uniqueness of strong solutions, and the exponential stability of solutions to (6) with initial and boundary value problem. For more results on viscoelastic equations, we can refer to Berrimi and Messaoudi , Messaoudi , Messaoudi and Tartar [19, 20], and the references therein.
In recent years, many mathematical workers studied some systems with time delay effects. Datko et al.  studied the following system:By using an observability inequality, they proved the exponential stability for the energy when . Subsequently, Xu et al.  obtained the same result as in  for the one space dimension by using the spectral analysis approach. Later on, Kirane and Said-Houari  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 and , they established the global well-posedness of the system. Moreover, under the assumption 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 and , one can refer to Nicaise and Pignotti , Nicaise et al. , Nicaise and Valein , 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 , 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 (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 . 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 , because the authors considered the case of a strong damping in .
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.
The notation in this paper will be as follows: , , , , , denote the usual (Sobolev) spaces on . In addition, denotes the norm in the space , and we also put .
2. Preliminaries and Main Results
In this section, we give some preparations for our consideration and our main results.(i)We assume that is a function satisfying (if is monotone nondecreasing).(ii)For the memory kernel , we assume that(G1) is a function satisfying where is the embedding constant for .(G2)There exists a positive nonincreasing differentiable function such that (iii)The nonlinear term satisfies where is a constant, and satisfiesWe denote and assume that
In order to deal with the delay feedback term, motivated by [24, 26], we introduce the following new dependent variable:Then it is easy to verifyThus, problem (1)–(3) is transformed intowith , and , and the initial and boundary conditions are
Let be a positive constant satisfying
Theorem 1. Let hold and assume the assumptions (8)–(14) hold.(i)If the initial data , , then problem (17)-(18) has a weak solution such that(ii)If the initial data , , where then the above weak solution has higher regularity(iii)In both cases, we have that the solution depends continuously on the initial data in . In particular, problem (17)-(18) has a unique weak solution.
3. 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.  and Jorge Silva and Ma .
3.1. Approximate Problem
Let be the Galerkin basis given by the eigenfunctions of with boundary condition on . For any , let .
We define for the sequence by Then we can extend by over and denote .
Given initial data , , and , we define the approximationswhich satisfy the following approximate problem: with initial conditions which satisfies
3.2. A Priori Estimate
Now multiplying the first approximate equation of (28) by , we see thatNoting the following fact:wherewe know that
Multiplying the second approximate equation of (28) by and then integrating over , we obtainA straightforward calculation givesNow integrating (34) and using (35)-(36) and , we infer thatwithThen we have the following cases.(i)Consider . Using Young’s inequality, we havewhich, together with (37), yieldsIt follows from (19) that there exist two constants and such that(ii)Consider . Taking and using (37), we know thatThen, in both cases, we infer that there exists a positive constant independent on such that It follows from (9), (14), and (43) thatThus we can obtain , for all .
3.3. Passage to Limit
From (44), we conclude that for any ,Thus we getBy (45)–(47), we can also deduce that is bounded in and is bounded in . Then from Aubin-Lions theorem , we infer that for any , We also obtain by Lemma 1.4 in Kim  that Then we can pass to limit the approximate problem (28)-(29) in order to get a weak solution of problem (17)-(18).
3.4. Continuous Dependence and Uniqueness
Let and () be two global solutions of problem (17)-(18) with respect to initial data and respectively. Let , . Then verifieswith boundary conditionsand initial dataMultiplying (47) by and integrating the result over , we getBy mean value theorem and Hölder’s inequality, we deriveIt follows from (12)-(13) and Hölder’s inequality thatMoreover,Noting that (35)-(36) and combining (54)–(57), we conclude thatwhereIt follows (19) thatwhich, along with (9), givesApplying Gronwall’s inequality to (61), we getThis shows that solution of problem (17)-(18) depends continuously on the initial data. In particular, problem (17)-(18) has a unique stronger solution.
We can prove the continuous dependence and uniqueness for weak solutions by using density arguments (see, e.g., Cavalcanti et al. ) which also can be found in Lions  (Chapter 1, Theorem 1.2) by using a regularization method and in Pata and Zucchi  or Giorgi et al.  by using the mollifiers.
This ends the proof of Theorem 1.
4. General Decay
We first consider stronger solutions. Define the modified energy bywhere is a positive constant satisfying (19).
Now we define the following functional:Then we have the following lemma.
Proof. By taking a derivative of (67) and using the first equation of (17), we conclude thatUsing Hölder’s inequality, we know that, for any ,By using Young’s inequality and Poincaré’s inequality and noting , we infer that, for any , Combining (69)–(72) and noting (14), we complete the proof.
In order to handle the term , we introduce the functional Then we have the following estimate.
Proof of Theorem 2. We define the Lyapunov functionalwhere is a real number which will be taken later.
First, we claim that there exist two positive constants and such that, for any , Indeed, it is easy to getwhere is the first eigenvalue of in with on . Choosing , we know thatNow putting small enough and choosing and , we see that (77) holds.
Next, combining (66), (68), and (74), we arrive atNow we choose and so small that we can take two positive constants and such that, for any , Multiplying (81) by , we have, for any , which, along with (10) and (66), impliesthat is,Denote , and then is equivalent to ; that is,Thus we conclude that, for any ,Integrating (86) over , we will see the following:which, together with (65), (77), and (85), gives (25).
This proves the general decay for regular solutions. We can extend the result to weak solutions by using a standard density argument; one can refer to Cavalcanti et al. . The proof is hence complete.
Remark 6. There are some open problems concerning our present work, and here we give some of them.(1)It is obvious that the weak damping term plays a crucial role in our proofs. It is still an open problem when .(2)We only obtain the general decay for . Whether the stability property holds for is still open.(3)It is interesting to study that the weight of the delay is bigger than the weight of the damping; that is, .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author would like to thank the referees for their helpful comments. This work was supported by the Fundamental Research Funds for the Central Universities with Contract no. JBK150128.
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