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

We consider a nonlinear viscoelastic wave equation utt t −k0Δu t ∫ t 0 g t− s div a x ∇u s ds b x ut f u , with nonlinear boundary damping in a bounded domain Ω. Under appropriate assumptions imposed on g 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.


Introduction
In this paper, we are concerned with the energy decay rate of the following viscoelastic problem with nonlinear boundary dissipation: where k 0 > 0 and Ω is a bounded domain in R n n ≥ 1 with a smooth boundary Γ Γ 0 ∪ Γ 1 .
Here, Γ 0 and Γ 1 are closed and disjoint with meas Γ 0 > 0, and ν is the unit outward normal 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 Ω is a bounded domain in R n n ≥ 1 with a smooth boundary, γ > 0, and a : Ω → R is a function, which may be null on a part of Ω.The authors established an exponential decay estimate under the conditions that a x ≥ a 0 > 0 on ω ⊂ Ω, with meas ω > 0 and satisfying some geometry conditions and −ξ 1 g t ≤ g t ≤ −ξ 2 g t , t ≥ 0. 1.4 Berrimi and Messaoudi 2 improved the result 1 by introducing a new function.They proved an exponential decay result under weaker conditions on both a and g.In fact, they allowed the function a to vanish on any part of Ω, 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:

1.5
Under some conditions on the relaxation function g, they improved the result of 1 .Indeed, they proved that the solution of 1.5 decays exponentially to zero when g is decaying exponentially and h is linear, and the solution decays polynomially to zero when g is decaying polynomially and h is nonlinear.
On considering the boundary stabilization, Cavalcanti et al. 6 considered the following problem: 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 b and g.In fact, we will allow the function b to be null on any part of Ω including Ω itself and the kernel function g 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.

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 H 1 Γ 0 with the Hilbert structure induced by H 1 Ω .We have that H 1 Γ 0 is a Hilbert space.For simplicity, we denote Let B > 0 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 B 1 , that is, Next, we state the assumptions for problem 1.1 as follows.
and there exists a nonincreasing positive differentiable function ξ such that for some positive constant α 1 .
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 .

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.

3.5
Next, we define a functional F, which helps in establishing the desired results.Setting 3.6 Remark 3.3.As in 17 , we can verify that the functional F is increasing in 0, λ 0 , decreasing in λ 0 , ∞ , and F has a maximum at λ 0 l p/2 /B p 1/ p−2 with the following maximum value:

3.7
Further, from 3.1 , 3.2 , 2.4 , and the definition of F by 3.6 , we have for all t ∈ 0, T .Moreover, one has J u t < E 1 and for all t ∈ 0, T .
Proof.Using 3.8 and considering E t is a nonincreasing function, we obtain Further, from Remark 3.3, we observe that F is increasing in 0, λ 0 , decreasing in λ 0 , ∞ , and
Next, we will prove that

3.13
To establish 3.13 , we argue by contradiction.Suppose that 3.13 does not hold, then there exists t * ∈ 0, T such that This contradicts 3.11 .

3.19
Therefore, we complete the proof of Lemma 3.4.

3.22
Thus, we have the inequality 3.20 and we also establish the boundedness of u t in L 2 Ω and the boundedness of u in H 1 Γ 0 .Moreover, from 2.2 and 3.22 , we also obtain the boundedness of u in L p Ω .Hence, it must have T ∞.
Additionally, using 2.2 and 3.20 , we obtain

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

3.30
Theorem 3.7.Let (A1)-(A3) and 1.2 hold.Assume that , and E 0 < E 1 .Then, for any t 0 > 0, there exist two positive constants K and k such that the solution of 1.1 satisfies 3.31 Proof.First, we estimate the derivative of G t .From 3.25 and using 1.1 , we have

3.32
The third, the fourth, and the fifth terms on the right-hand side of 3.32 can be estimated as follows.From H ölder's inequality, Young's inequality, and 2.4 , for η > 0, we have

3.36
Next, we estimate Ψ t .Taking the derivative of Ψ t in 3.26 and using 1.1 to obtain

3.37
As in deriving 3.36 , in what follows we will estimate the right-hand side of 3.37 .Using Young's inequality, H ölder's inequality, 2.4 , and 2.9 , for δ > 0, we have

3.39
Utilizing H ölder's inequality, Young's inequality, 2.3 , and 2.7 , the sixth term on the righthand side of 3.37 can be estimated as

3.40
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

3.41
Journal of Function Spaces and Applications 13 Combining these estimates 3.38 -3.41 , 3.37 becomes 4a 0 δ .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 t 0 > 0, t 0 g s ds ≥ t 0 0 g s ds g 0 , ∀t ≥ t 0 , 3.44 because g is positive and continuous with g 0 > 0. At this point, we choose δ > 0 small enough so that 4δc 5 l < a 0 g 0 2 < a 0 g 0 − δ.

3.45
Whence δ is fixed, the choice of any two positive constants ε 1 and ε 2 satisfying

3.47
Then, we choose ε 1 and ε 2 so small that 3.27 and 3.45 remain valid, further

3.48
Hence, for all t ≥ t 0 , we arrive at The existence and uniform decay rate results were established under quite restrictive assumptions on damping term h and the kernel function g.Later, Cavalcanti et al. 7 generalized this result without imposing a growth condition on h and under a weaker assumption on g.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 h and greatly weakened the assumption on g.Very recently, problem 1.1 has been considered by Li et al. 10 with b x 0 and f u −|u| γ u, γ > 0. 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 g.We refer the reader to related works 7, 11-16 dealing with boundary stabilization.
− s |∇u s − ∇u t | 2 |∇u t | 2 ds dx k 2 u t needed, one can choose ε 1 sufficiently small G t ≤ −c 9 E t c 10 g • ∇u t , 3.50 where c i , i 7, 8, 9, are some positive constants.It follows from 3.50 , 2.5 , and 3.5 that 2c 10 E t is equivalent to E t by Lemma 3.6 and k is a positive constant.A integration of 3.52 leads to L t ≤ L t 0 e Remark 3.8.We illustrate the energy decay rate given by Theorem 3.7 through the following examples which are introduced in 18 .0 and ν > 1 and α 1 ∈ R or ν 1 and α 1 > 1 , then for ξ t G t ≤ −c 9 ξ t E t c 10 ξ t g • ∇u t ≤ −c 9 ξ t E t − c 10 g • ∇u t ≤ −c 9 ξ t E t − 2c 10 E t , for t ≥ t 0 .L t ≤ −c 9 ξ t E t ≤ −kξ t L t , for t ≥ t 0 ,where K is a positive constant.This completes the proof.