Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 421847, 17 pages

doi:10.1155/2012/421847

Research Article

Uniform Decay of Solutions for a Nonlinear Viscoelastic Wave Equation with Boundary Dissipation

Shun-Tang Wu¹ and Hsueh-Fang Chen²

¹General Education Center, National Taipei University of Technology, Taipei 106, Taiwan

²Department of Business Administration, Yu Da University, Miaoli 366, Taiwan

Received 27 June 2012; Accepted 11 August 2012

Academic Editor: Pankaj Jain

Copyright © 2012 Shun-Tang Wu and Hsueh-Fang Chen. 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.

We consider a nonlinear viscoelastic wave equation , with nonlinear boundary damping in a bounded domain . Under appropriate assumptions imposed on and with certain initial data, we establish the general decay rate of the solution energy which is not necessarily of exponential or polynomial type. This work generalizes and improves earlier results in the literature.

1. Introduction

In this paper, we are concerned with the energy decay rate of the following viscoelastic problem with nonlinear boundary dissipation:

where and is a bounded domain in with a smooth boundary . Here, and are closed and disjoint with , and is the unit outward normal to . The relaxation function is a positive and uniformly decaying function, and are functions satisfying some conditions given in (A2) and (A3), respectively, is a function, and with

This type of equations usually arise in the theory of viscoelasticity. It is well known that viscoelastic materials have memory effects, which is due to the mechanical response influenced by the history of the materials themselves. As these materials have a wide application in the natural sciences, their dynamics are interesting and of great importance. From the mathematical point of view, their memory effects are modeled by integrodifferential equations. Hence, questions related to the behavior of the solutions for the PDE system have attracted considerable attention in recent years.

For example, Cavalcanti et al. [1] considered the following problem:

where is a bounded domain in with a smooth boundary, , and is a function, which may be null on a part of . The authors established an exponential decay estimate under the conditions that on , with and satisfying some geometry conditions and

Berrimi and Messaoudi [2] improved the result [1] by introducing a new function. They proved an exponential decay result under weaker conditions on both and . In fact, they allowed the function to vanish on any part of , and, consequently, the geometry condition imposed on a part of boundary is no longer needed. Later, the same authors [3] and Messaoudi [4] extended the result to a situation in which a source term is competing with the viscoelastic dissipation. In [5], Cavalcanti and Oquendo considered the following:

Under some conditions on the relaxation function , they improved the result of [1]. Indeed, they proved that the solution of (1.5) decays exponentially to zero when is decaying exponentially and is linear, and the solution decays polynomially to zero when is decaying polynomially and is nonlinear.

On considering the boundary stabilization, Cavalcanti et al. [6] considered the following problem:

The existence and uniform decay rate results were established under quite restrictive assumptions on damping term and the kernel function . Later, Cavalcanti et al. [7] generalized this result without imposing a growth condition on and under a weaker assumption on . Recently, Messaoudi and Mustafa [8] exploited some properties of convex functions [9] and the multiplier method to extend these results. They established an explicit and general decay rate result without imposing any restrictive growth assumption on the damping term and greatly weakened the assumption on . Very recently, problem (1.1) has been considered by Li et al. [10] with and . They showed the global existence and uniqueness of global solution of problem (1.1) and established uniform decay rate of the energy under suitable conditions on the initial data and the relaxation function . We refer the reader to related works [7, 11–16] dealing with boundary stabilization.

Motivated by previous works, it is interesting to investigate the global existence and uniform decay result of solutions to problem (1.1) when a forcing source term is competing with the viscoelastic dissipation and nonlinear boundary damping under the weaker assumption on both and . In fact, we will allow the function to be null on any part of (including itself) and the kernel function is not necessarily decaying in an exponential or polynomial fashion. Therefore, our result allows a larger class of relaxation functions and improves the results in [10, 13] where only the exponential and polynomial rate was considered.

The remainder of this paper is organized as follows. In Section 2, we provide assumptions that will be used later and mention the local existence result Theorem 2.1. In Section 3, we prove our stability result that is given in Theorem 3.7.

2. Preliminary Results

In this section, we give assumptions and preliminaries that will be needed throughout the paper. First, we introduce the following set:

and endow with the Hilbert structure induced by . We have that is a Hilbert space. For simplicity, we denote and . According to (1.2), we have the imbedding: . Let be the optimal constant of Sobolev imbedding which satisfies the following inequality:

and we use the Trace-Sobolev imbedding: . In this case, the imbedding constant is denoted by , that is,

Next, we state the assumptions for problem (1.1) as follows.

(A1) is a bounded function satisfying

and there exists a nonincreasing positive differentiable function such that

(A2) is a nondecreasing function with

(A3) is a nonnegative functions and such that

for some positive constant .

By using the Galerkin method and procedure similar to that of [10, 16], we can have the following local existence result for problem (1.1).

Theorem 2.1. Let hypotheses (A1)–(A3) and (1.2) hold and assume that . Then there exists a strong solution of (1.1) satisfying

Furthermore, if , then there exists a weak solution of (1.1) satisfying

for some .

3. Global Existence and Energy Decay

In this section, we focus our attention on the uniform decay of weak solutions to problem (1.1). For this purpose, we define

and the energy function

where

Adopting the proof of [10], we still have the following results.

Lemma 3.1. For any , we have

Lemma 3.2. Let be the solution of (1.1), then, under assumptions (A1)-(A2), is a nonincreasing function on and

Next, we define a functional , which helps in establishing the desired results. Setting

Remark 3.3. As in [17], we can verify that the functional is increasing in , decreasing in , and has a maximum at with the following maximum value:

Further, from (3.1), (3.2), (2.4), and the definition of by (3.6), we have

Lemma 3.4. Suppose that (A1)-(A2) and (1.2) hold. Assume further that , and satisfy and . Then, it holds that

for all . Moreover, one has and

for all .

Proof. Using (3.8) and considering is a nonincreasing function, we obtain

Further, from Remark 3.3, we observe that is increasing in , decreasing in , and as . Thus, as , there exist such that , which together with infer that

This impliesthat .

Next, we will prove that

To establish (3.13), we argue by contradiction. Suppose that (3.13) does not hold, then there exists such that

Case 1. If , then

This contradicts (3.11).

Case 2. If , then by continuity of , there exists such that

then

This is also a contradiction of (3.11). Thus, we have proved the inequality (3.13).

To prove (3.10), we note for , such that

because of . Thanks to by (3.9), we obtain

Therefore, we complete the proof of Lemma 3.4.

Theorem 3.5. Let , and (A1)-(A2) and (1.2) hold. Assume further that and , then the problem (1.1) admits a global solution. Furthermore, for all , one has

with .

Proof. It follows from (3.19) and (3.8) that

Thus, we have the inequality (3.20) and we also establish the boundedness of in and the boundedness of in . Moreover, from (2.2) and (3.22), we also obtain the boundedness of in . Hence, it must have .

Additionally, using (2.2) and (3.20), we obtain

for all .

Now, we will investigate the asymptotic behavior of the energy function . First, we define some functionals and establish Lemma 3.6. Let

where

and , are some positive constants to be be specified later.

Lemma 3.6. There exist two positive constants and such that the relation

holds, for small enough.

Proof. By Hölder’s inequality, Young’s inequality, (2.2), and (2.8), we deduce that

Hence, taking (3.24) and (3.28) into account, we have

where , and . Thus, using (3.22) and selecting small enough, there exist two positive constants and such that

Theorem 3.7. Let (A1)–(A3) and (1.2) hold. Assume that , and . Then, for any , there exist two positive constants and such that the solution of (1.1) satisfies

Proof. First, we estimate the derivative of . From (3.25) and using (1.1), we have

The third, the fourth, and the fifth terms on the right-hand side of (3.32) can be estimated as follows. From Hölder’s inequality, Young’s inequality, and (2.4), for , we have

Employing Hölder’s inequality, Young’s inequality, (2.2), (2.3), and (2.7), for , we see that

A substitution of (3.33)-(3.34) into (3.32) yields

Letting and in the above inequality, we obtain

Next, we estimate . Taking the derivative of in (3.26) and using (1.1) to obtain

As in deriving (3.36), in what follows we will estimate the right-hand side of (3.37). Using Young’s inequality, Hölder’s inequality, (2.4), and (2.9), for , we have

Again, exploiting (2.9), Young’s inequality, Hölder’s inequality, and (2.4), we obtain

Utilizing Hölder’s inequality, Young’s inequality, (2.3), and (2.7), the sixth term on the right-hand side of (3.37) can be estimated as

As for the seventh and the eighth terms on the right-hand side of (3.37), using Hölder’s inequality, Young’s inequality, (2.2), (3.20), and (2.4), we obtain

Combining these estimates (3.38)–(3.41), (3.37) becomes

where and . Hence, we conclude from (3.24), (3.5), (3.36), (3.42), and (2.6) that

where we have used the fact that for any ,

because is positive and continuous with . At this point, we choose small enough so that

Whence is fixed, the choice of any two positive constants and satisfying

will make

Then, we choose and so small that (3.27) and (3.45) remain valid, further

Hence, for all , we arrive at

which yields (if needed, one can choose sufficiently small)

where are some positive constants. It follows from (3.50), (2.5), and (3.5) that

That is,

where is equivalent to by Lemma 3.6 and is a positive constant. A integration of (3.52) leads to

Again, employing is equivalent to leads to

where is a positive constant. This completes the proof.

Remark 3.8. We illustrate the energy decay rate given by Theorem 3.7 through the following examples which are introduced in [18].

(i) If

then (3.54) gives the exponential decay estimate

Similarly, if

then we obtain the polynomial decay estimate

(ii) If

with , then (2.5) holds for

Thus (3.54) gives the estimate

(iii) If

where and and , then for

we obtain from (3.54) that

Acknowledgment

The authors would like to thank very much the referees for their important remarks and comments which allow them to correct and improve this paper.

References

M. M. Cavalcanti, V. N. Domingos Cavalcanti, and J. A. Soriano, “Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping,” Electronic Journal of Differential Equations, vol. 44, pp. 1–14, 2002.

S. Berrimi and S. A. Messaoudi, “Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping,” Electronic Journal of Differential Equations, vol. 2004, no. 88, pp. 1–10, 2004.

S. Berrimi and S. A. Messaoudi, “Existence and decay of solutions of a viscoelastic equation with a nonlinear source,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 10, pp. 2314–2331, 2006.

S. A. Messaoudi, “General decay of the solution energy in a viscoelastic equation with a nonlinear source,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2589–2598, 2008.

M. M. Cavalcanti and H. P. Oquendo, “Frictional versus viscoelastic damping in a semilinear wave equation,” SIAM Journal on Control and Optimization, vol. 42, no. 4, pp. 1310–1324, 2003.

M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho, and J. A. Soriano, “Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping,” Differential and Integral Equations, vol. 14, no. 1, pp. 85–116, 2001.

M. M. Cavalcanti, V. N. Domingos Cavalcanti, and P. Martinez, “General decay rate estimates for viscoelastic dissipative systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 1, pp. 177–193, 2008.

S. A. Messaoudi and M. I. Mustafa, “On convexity for energy decay rates of a viscoelastic equation with boundary feedback,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 9-10, pp. 3602–3611, 2010.

F. Alabau-Boussouira, “Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems,” Applied Mathematics and Optimization, vol. 51, no. 1, pp. 61–105, 2005.

F. Li, Z. Zhao, and Y. Chen, “Global existence uniqueness and decay estimates for nonlinear viscoelastic wave equation with boundary dissipation,” Nonlinear Analysis: Real World Applications, vol. 12, no. 3, pp. 1759–1773, 2011.

W. J. Liu and J. Yu, “On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 6, pp. 2175–2190, 2011.

L. Lu, S. Li, and S. Chai, “On a viscoelastic equation with nonlinear boundary damping and source terms: global existence and decay of the solution,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 295–303, 2011.

F. Li and C. Zhao, “Uniform energy decay rates for nonlinear viscoelastic wave equation with nonlocal boundary damping,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 11, pp. 3468–3477, 2011.

S. A. Messaoudi and A. Soufyane, “General decay of solutions of a wave equation with a boundary control of memory type,” Nonlinear Analysis: Real World Applications, vol. 11, no. 4, pp. 2896–2904, 2010.

J. E. Muñoz Rivera and A. Peres Salvatierra, “Asymptotic behaviour of the energy in partially viscoelastic materials,” Quarterly of Applied Mathematics, vol. 59, no. 3, pp. 557–578, 2001.

J. Y. Park and S. H. Park, “Existence and asymptotic stability for viscoelastic problems with nonlocal boundary dissipation,” Czechoslovak Mathematical Journal, vol. 56, no. 2, pp. 273–286, 2006.

E. Vitillaro, “Global existence for the wave equation with nonlinear boundary damping and source terms,” Journal of Differential Equations, vol. 186, no. 1, pp. 259–298, 2002.

S. A. Messaoudi, “General decay of solutions of a viscoelastic equation,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 1457–1467, 2008.