Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 585021, 10 pages
http://dx.doi.org/10.1155/2015/585021

Research Article

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

Baowei Feng

College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Received 21 January 2015; Revised 18 March 2015; Accepted 18 March 2015

Academic Editor: Ivanka Stamova

Copyright © 2015 Baowei Feng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

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 [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. A more general equation of (4) reads

where 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 [2], Cavalcanti et al. [3, 4], Ma [5], Ma and Narciso [6], de Lacerda Oliveira and de Lima [7], J. Y. Park and S. H. Park [8], Patcheu [9], Rivera [10, 11], Tusnal [12], Vasconcellos and Teixeira [13], Yang [14, 15], and the references therein. Very recently, Andrade et al. [16] investigated a viscoelastic plate equation with -Laplacian and memory terms with strong damping

where 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 [17], Messaoudi [18], 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. [21] studied the following system:

By using an observability inequality, they proved the exponential stability for the energy when . 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 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 [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 , 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 [16], because the authors considered the case of a strong damping 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.

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 satisfies

We 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 verify

Thus, problem (1)–(3) is transformed into

with , and , and the initial and boundary conditions are

Let be a positive constant satisfying

Now we define the weak solutions of (1)–(3): for given initial data , we say that a function is a weak solution to the problem (1)–(3) if and

for all .

Next we state the global well-posedness of problem (17)-(18) given in the following theorem.

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.

We define the energy of problem (17)-(18) by

Finally, we give the energy decay of problem (17)-(18).

Theorem 2. Let hold and assume the assumptions (8)–(14) hold. In both cases (i) and (ii), there exist two constants and such that the energy defined by (24) satisfies

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. [16] and Jorge Silva and Ma [28].

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 approximations

which satisfy the following approximate problem:

with initial conditions

which satisfies

By using standard ordinary differential equations theory, the problem (28)-(29) has a solution defined on . The following estimate will give the local solution being extended to , for any given .

3.2. A Priori Estimate

Now multiplying the first approximate equation of (28) by , we see that

Noting the following fact:

where

we know that

Multiplying the second approximate equation of (28) by and then integrating over , we obtain

A straightforward calculation gives

Now integrating (34) and using (35)-(36) and , we infer that

with

Then we have the following cases.

(i)Consider  . Using Young’s inequality, we have

which, together with (37), yields

It follows from (19) that there exist two constants and such that

(ii)Consider  . Taking and using (37), we know that

Then, in both cases, we infer that there exists a positive constant independent on such that

It follows from (9), (14), and (43) that

Thus we can obtain , for all .

3.3. Passage to Limit

From (44), we conclude that for any ,

Thus we get

By (45)–(47), we can also deduce that is bounded in and is bounded in . Then from Aubin-Lions theorem [29], we infer that for any ,

We also obtain by Lemma 1.4 in Kim [30] 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

Firstly we prove the continuous dependence and uniqueness for stronger solutions of problem (17)-(18).

Let and () be two global solutions of problem (17)-(18) with respect to initial data and respectively. Let , . Then verifies

with boundary conditions

and initial data

Multiplying (47) by and integrating the result over , we get

By mean value theorem and Hölder’s inequality, we derive

It follows from (12)-(13) and Hölder’s inequality that

Moreover,

Noting that (35)-(36) and combining (54)–(57), we conclude that

where

It follows (19) that

which, along with (9), gives

Applying Gronwall’s inequality to (61), we get

This 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. [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.

4. General Decay

In this section, we will establish the decay property of the solution for problem (17)-(18) in the case . 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,

Lemma 3. Under the assumptions in Theorem 2, the modified energy functional defined by (63) satisfies that there exists a constant such that, for any ,

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 defined in (67) satisfies that, for any ,

where is the Poincaré constant.

Proof. By taking a derivative of (67) and using the first equation of (17), we conclude that

Using 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.

Lemma 5. Under the assumptions in Theorem 2, the functional defined in (73) satisfies that

where is a positive constant.

Proof. Differentiating (73) with respect to and using the second equation (17), we obtain

Then it is easy to verify that there exists a constant satisfying (74).

Proof of Theorem 2. We define the Lyapunov functional

where 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 get

where is the first eigenvalue of in with on . Choosing , we know that

Now putting small enough and choosing and , we see that (77) holds.
Next, combining (66), (68), and (74), we arrive at

Now 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), implies

that 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. [27]. 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.

Acknowledgments

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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