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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 102594, 14 pages
Boundary Stabilization of a Nonlinear Viscoelastic Equation with Interior Time-Varying Delay and Nonlinear Dissipative Boundary Feedback
1College of Science, National University of Defense Technology, Changsha, Hunan 410083, China
2School of Mathematics, Hunan Institute of Science and Technology, Yueyang, Hunan 414006, China
3School of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, China
Received 22 January 2014; Revised 2 May 2014; Accepted 16 May 2014; Published 9 June 2014
Academic Editor: Chuangxia Huang
Copyright © 2014 Zaiyun Zhang et al. 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.
We investigate a nonlinear viscoelastic equation with interior time-varying delay and nonlinear dissipative boundary feedback. Under suitable assumptions on the relaxation function and time-varying delay effect together with nonlinear dissipative boundary feedback, we prove the global existence of weak solutions and asymptotic behavior of the energy by using the Faedo-Galerkin method and the perturbed energy method, respectively. This result improves earlier ones in the literature, such as Kirane and Said-Houari (2011) and Ammari et al. (2010). Moreover, we give an positive answer to the open problem given by Kirane and Said-Houari (2011).
In this paper, we consider the global existence and asymptotic behavior of a nonlinear viscoelastic equation with interior time-varying delay and nonlinear dissipative boundary feedback as follows: where is a bounded domain of with a smooth boundary of is a positive real constant, represents the time-varying delay effect and the initial data are given functions belonging to suitable spaces, is a positive function that represents the kernel of the memory term, is nonlinear dissipative boundary feedback, and satisfy suitable assumptions (see in Section 2).
This model appears in viscoelasticity (see [1, 2]). In the case of velocity-dependent material density (i.e., ) as well as presence of and in the absence of the memory effect (i.e., ), (1) reduces to the wave equation. There is large literature on the global existence and uniform stabilization of wave equations. We refer the readers to [3–5]. It is worth mentioning that Zhang and Miao  considered the nonlinear wave equation with dissipative term and boundary damping and they proved the existence and uniform decay of strong and weak solutions by using the Glerkin method and the multiplier technique, respectively. Later on, Zhang et al.  improved earlier ones in . More precisely, they investigated the global existence and uniform stabilization of generalized dissipative Klein-Gordon equation with boundary damping and they proved the existence and uniform decay of strong and weak solutions by using the nonlinear semigroup method, the perturbed energy method, and the multiplier technique. Quite recently, Cavalcanti et al.  considered the following model: where is a smooth oriented embedded compact surface without boundary in and is the Laplace-Beltrami operator on manifold ; furthermore, they obtained explicit and optimal decay rates of the energy. Later on, Cavalcanti et al.  extended the result for n-dimensional compact Riemannian manifolds with boundary in two ways: (i) by reducing arbitrarily the region where the dissipative effect lies (this gives us a totally sharp result with respect to the boundary measure and interior measure where the damping is effective) and (ii) by controlling the existence of subsets on the manifold that can be left without any dissipative mechanism, namely, a precise part of radially symmetric subsets. An analogous result holds for compact Riemannian manifolds without boundary.
In the case and in the absence of delay (i.e., ), there is large literature on the existence and decay of nonlinear viscoelastic equation during the past decades. In , Cavalcanti et al. considered the exponential decay for the solution of viscoelastic wave equation with localized damping Under the condition that on , with satisfying some geometry restrictions and
they proved an exponential decay result for the energy. Berrimi and Messaoudi  improved Cavalcanti’s result by introducing a differential functional which allowed to weaken the conditions on both and . In , Cavalcanti and Oquendo studied Under some geometric restrictions on and assuming that they established an exponential stability for the relaxation function decaying exponentially and linear and polynomial stability for decaying polynomially and nonlinear. It is worth mentioning that Zhang et al.  studied the following initial boundary value problem: Furthermore, they showed that the solutions of (9) decay uniformly in time, with rates depending on the rate of decay of the kernel . More precisely, the solution decays exponentially to zero provided that decays exponentially to zero. When decays polynomially, we show that the corresponding solution also decays polynomially to zero with the same rate of decay. For other related works, we refer the readers to [12–21] and the references therein.
On the other hand, concerning the study of the following nonlinear viscoelastic equation with memory, there are a substantial number of contributions: Recently, Han and Wang  investigated the following problem: By introducing a new functional and using potential well method, the authors established the global existence and uniform decay if the initial data are in a suitable stable set. Cavalcanti et al.  studied a related problem with strong damping as follows: By assuming , if or and if and decays exponentially, they established that the global existence resulted for and the exponential decay of the energy for . This result has been extended to a situation by Messaoudi and Tatar  and exponential decay and polynomial decay results have been shown in the absence as well as presence of a source term. Later on, inspired by the ideas of [25–27], Han and Wang  investigated the general decay of solutions of energy for the nonlinear viscoelastic equation
In recent years, the control of partial differential equation with time delay effects has become an active area of research; see, for instance, [28, 29] and the references therein. The presence of delay may be a source of instability. For instance, it was proved in [30–34] that an arbitrarily small delay may destabilize a system which is uniformly asymptotically stable in the absence of delay unless additional conditions or control terms have been used. In , Nicaise and Pignotti examined (1) with , and being a constant delay in the case of mixed homogeneous Dirichlet-Neumann boundary conditions, under a geometric condition on the Neumann part of the boundary. More precisely, they investigated the following system with linear frictional damping term and internal constant delay: or with boundary constant delay In the presence of delay (), Nicaise and Pignotti  examined systems (14) and (15) and proved under the assumptions that the energy is exponentially stable. Otherwise, they constructed a sequence of delays for which the corresponding solution is instable. The main approach used there is an observability inequality together with a Carleman estimate. See also  for treatment to these problems in more general abstract form and  for analogous results in the case of boundary time-varying delay. We also recall the result by Nicaise et al. , where the researchers proved the same result as in  for the one space dimension by applying the spectral analysis approach. Recently, Kirane and Said-Houari  considered (1) with , and being a constant delay in the case of the initial and Dirichlet boundary wave equation with a linear damping and a delay term as follows: Under an assumption between the weight of the delay term in the feedback and the weight of the term without delay, using the Faedo-Galerkin method combined with some energy estimate, they proved the global existence of (16). Also, they proved exponential decay of (16) via suitable Lyapunov functionals.
Recently, the stability of PDEs with time-varying delays was studied in [38–44]. In , Nicaise and Pignotti investigated the stabilization problem by interior damping of the wave equation with internal time-varying delay feedback and established exponential stability estimates by introducing suitable Lyapunov functionals, under the condition in which the positivity of the coefficient is not necessary. In , Nicaise et al. showed the exponential stability of the heat and wave equations with time-varying boundary delay in 1-D, under the condition , where is a constant such that .
The rest of the paper is organized as follows. In Section 2, we show some assumptions and state our main result. In Section 3, we present the proof of our main result. That is, we will prove the global existence by using Faedo-Galerkin method and establish the general decay result (including exponential decay and polynomial decay) by using the perturbed energy method. Finally, in Section 4, we give further remarks on this context.
2. Some Assumptions and Main Results
In this section, before proceeding to our analysis, we present some assumptions and state the main result. We use the standard Hilbert space and the Sobolev space with their usual scalar products and norms. Throughout this paper, is used to denote a generic positive constant from line to line.
For the relaxation function , we assume that(G1) is a nonincreasing differentiable function such that (G2)there exists a nonincreasing differentiable function such that We assume that satisfies For the time-varying delay, we assume that there exist positive constant such that Furthermore, we assume that the delay satisfies that and that satisfy
Remark 1. We show an example of functions satisfying (G2) as follows: for to be chosen properly; see .
Remark 2. Condition is imposed so that .
Now, we are in a position to state our main results.
Theorem 3. Let (20)–(23) be satisfied and satisfy (G2). Then, given , and , there exists a unique weak solution such that Moreover, if (20)–(23) hold and satisfies (G1) and (G2), then there exist two positive constants such that for any solution of the problem (1) of the energy satisfies
3. Proof of the Main Result
In this section, we will divide our proof into two steps. In Step 1, we prove the global existence of weak solutions by using Faedo-Galerkin method benefited from the ideas of [2, 3, 37]. In Step 2, we establish the general decay of energy by introducing the new energy functional and using the perturbed energy method inspired by the contributions; see, for instance, [2–4, 11, 39].
Step 1 (global existence of weak solutions). Let be an orthogonal basis of with being the eigenfunction of the following problem: Denote for subspace generated by the first vectors of the basis of . Then, we construct approximation of the solution as follows: and we choose two sequences and in and a sequence in such that strongly in strongly in , and strongly in . Define the sequence as follows: . Then, from [37, pp 1069], we may extend by over and denote .
To facilitate further our analysis, we introduce as in [32, 36, 39] the new variable Then, we get Therefore, the problem (1) can be rewritten as follows: Hence, are solutions to the following Cauchy problem as follows: By standard method of ODE, we know that there exists only one local solution of the Cauchy problem (33) and (34) on some interval , for arbitrary ; then, this solution can be extended to the whole interval by a priori estimates below.
Lemma 4. For and , one has
Lemma 6. Assuming that is a continuous function such that Then, we have
Proof. It suffices to observe that, for , By applying Hölder inequality, we obtain Taking , we get Finally, taking in the above equality, Lemma 6 is completed.
3.1. A Priori Estimate
Taking in (33) and integrating over , using integration by parts and Lemma 4, we obtain Taking in (34) and integrating over , we get Now, integrating by parts, we obtain It follows from (43) and (44) that Summing up (42) and (45), we conclude that where Using Young’s inequality and noticing (20) and (21), we arrive at Choosing some value of and and noticing (20) and (21), we have . Moreover, choosing some value of and , we obtain That is, In fact, by (20) and (21), we get . From (48) and (50), (G1), and (G1) and Lemma 6, we conclude that we can find a positive independent of , such that Hence, using the fact that , the estimate (51), and equality (47), we deduce By (52), we infer that there exist two subsequences (still denoted by ) and two functions and , such that From (52), we have is bounded in and is bounded in . Consequently, is bounded in . More details are present in [37, pp 1072].
Since the Sobolev embedding is compact, using Aubin-Lions theorem (see ), we can extract a subsequence of (still denoted by ), such that which implies almost everywhere in .
Hence, On the other hand, by the Sobolev embedding theorem and estimate (51), this yields where is the Sobolev embedding constant. Thus, using (55), (56), and Lions Lemma , we get Let be the space of functions with compact support in . Multiplying the first equation in (33) by and integrating over , we conclude that Noticing that is a basis of , via convergence (53) and (57), we can pass to the limit in (58) and obtain Similarly, we get From (53) and given the label of lemma in , we obtain Therefore, we have . Consequently, the global existence of weak solution is established.
Step 2 (general decay of the energy). First, we introduce the new energy functional and the perturbed energy ; then we apply the perturbed energy method to establish general decay of the energy. More precisely, the method used is based on the construction of suitable Lyapunov functionals and satisfying for some positive constants . More details are present in [3, pp 1017] or [2, 4, 16].
Now, we introduce the new energy functional as follows: where are suitable positive constants.
Next, we will fix such that
Remark 7. In fact, the existence of such a constant is guaranteed by the assumption (23).
Therefore, we have the following lemma.
Proof of Lemma 8. Differentiating (63) and noticing the first equation in (1) together with we obtain Applying Young’s inequality, we obtain Integrating by parts, using the assumption (20), (21) and (67), (68), we arrive at Combining (64) and (69) and the assumptions (G1) and (G2), (65) is established.
Next, we introduce the following functionals: Set where and are suitable positive constants to be determined later.
Remark 9. Indeed, we easily see that, for small enough while large enough, there exist two positive constants , such that Concerning the estimates of , we have the following lemmas.
Lemma 10. Under the assumption (G1), the functional satisfies the estimate
Proof of Lemma 10. Differentiating (70) and integrating by parts, we get Using Young’s inequality and (G1), we obtain (see ) Also, applying Young’s and Poincaré’s inequality yields Noticing (75)–(77) and choosing small enough, we obtain estimate (74).
Lemma 11. Under the assumption (G1), the functional satisfies the estimate
Proof of Lemma 11. Differentiating (71), integrating by parts, and noticing the first equation in (1), we have Observe that It follows from (79) and (80) that Using Young’s and Poincaré’s inequality, we get (see ) From (81) and (82), we derive Lemma 11.