Research Article | Open Access
Shun-Tang Wu, Hsueh-Fang Chen, "Uniform Decay of Solutions for a Nonlinear Viscoelastic Wave Equation with Boundary Dissipation", Journal of Function Spaces, vol. 2012, Article ID 421847, 17 pages, 2012. https://doi.org/10.1155/2012/421847
Uniform Decay of Solutions for a Nonlinear Viscoelastic Wave Equation with Boundary Dissipation
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.
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.  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  improved the result  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  and Messaoudi  extended the result to a situation in which a source term is competing with the viscoelastic dissipation. In , Cavalcanti and Oquendo considered the following: Under some conditions on the relaxation function , they improved the result of . 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.  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.  generalized this result without imposing a growth condition on and under a weaker assumption on . Recently, Messaoudi and Mustafa  exploited some properties of convex functions  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.  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 .
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 , 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 , 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.
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
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 .(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
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.
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