Uniform decay estimates for the semi - linear wave equation with locally distributed mixed - type damping via arbitrary local viscoelastic versus frictional dissipative

: We are concerned with the stabilization of the wave equation with locally distributed mixed - type damping via arbitrary local viscoelastic and frictional e ﬀ ects. Here, one of the novelties is: the viscoelastic and frictional damping together e ﬀ ect only in a part of domain, not in entire domain, which is only assumed to meet the piecewise multiplier geometric condition that their summed interior and boundary measures can be arbi - trarily small. Furthermore, there is no other additional restriction for the location of the viscoelastic - e ﬀ ect region. That is, it is dropped that the viscoelastic - e ﬀ ect region includes a part of the system boundary, which is the fundamental condition in almost all previous literature even if when two types of damping together cover the entire system domain. The other distinct novelty is: in this article we remove the fundamental condition that the derivative of the relaxation function is controlled by relaxation function itself, which is a necessity in the previous literature to obtain the optimal uniform decay rate. Under such weak conditions, we successfully establish a series of decay theorems, which generalize and extend essentially the previous related stability results for viscoelastic model regardless of local damping case, entire damping case and mixed - type damping case.


Introduction
Let Ω be a bounded connected domain of R n with smooth boundary Ω ∂ .Consider the following semi-linear wave equation subjected to locally distributed mixed-type damping-viscoelastic and frictional dissipative effects: where g t ( ) is the relaxation function, which is monotone non-increasing and produces viscoelastic dissipation in (1.1); the function h corresponds to the frictional damping.The coefficients a x b x , 0 ( ) ( ) ≥ , which only depend on the spatial variables, are responsible for the effectiveness of each damping mechanism on a part of Ω where a x b x 0 ( ) ( ) + > , not on the entire domain Ω.Thus, problem (1.1) is a local mixed-type damping system.Next, we will investigate the following issues deeply in this article: • What is the asymptotic behavior of system (1.1) with only local damping, and is it possible to achieve the same decay rates as entire damping case?Or for local damping case, what kind of geometric control condition can produce the same asymptotic behavior as entire damping case?• How to obtain the uniform stability for system (1.1) with restrictions as small as possible on the kernel function g and the coefficients a x b x , ( ) ( )?
For this purpose, the aim of this article is to study the decay properties and asymptotic behavior associated with the local damping problem (1.1).The focuses are on analyzing the effects of each type of damping and establishing some decay results in a general framework.
If b x 0 ( ) = in (1.1), i.e., u u g t s a x u s s Δ d i v d 0 , i nΩ 0 , .
(1.3) is a damping wave system only with viscoelastic term g t s a x u s s div d ∫ − ∇ , but no frictional term.When a x 0 ( ) > holds in entire Ω, thanks to the work of Dafermos [18], Dafermos and Nokel [20], the studies of well-posedness and stability for system (1.3) have been made for nonnegative, decreasing, smooth relaxation functions g with some control conditions on g′ in lots of literature (cf., e.g., [4,8,37] and references therein).However, when the coefficient a x 0 ( ) > holds in a part of Ω, it was only studied in [48] (to the best of our knowledge) under some special conditions on g and a.
Moreover, many other related wave equations or systems have been significantly studied and a large variety of work has been obtained (cf., e.g., [6,8,[13][14][15][16][17]19,21,22,[24][25][26]30,32,35,39,43,45,47,48,51,54,55] and references therein).For example, the abstract coupled system, i.e., In summary, first, the previous research mainly focused on wave equation (1.2) with locally frictional damping, and the support of b x ( ) satisfies the usual multiplier geometric condition (Γ-condition); second, there are few results on wave equation (1.3) with locally viscoelastic damping, and the support of a x ( ) must also satisfy the usual multiplier geometric condition; third, for mixed-type damping case, the existing results mainly focused on wave equation with complementary viscoelastic and frictional damping in entire domain.To the best of our knowledge, there is no literature to study (1.1) with arbitrary local viscoelastic versus frictional damping.
Besides, due to the localization of the damping in (1.2) and (1.3), the a supp( ) or b supp( ) must satisfy some geometric conditions (see [36,48,56]) in order to ensure the decay properties of the whole energy.Therefore, it seems natural that we shall have to impose some geometric conditions on the localization of the damping terms in (1.1), i.e., on the support of the function a x b x ( ) ( ) + .Furthermore, in order to obtain uniform (or optimal) stability, many different types of control conditions on the derivative g′ of relaxation function g were introduced and used in previous literature, such as, for t 0 ≥ , g t ξ t g t g t g t g t H g t ; or ; or .
In this article, our main contributions are as follows: • The problem studied here is about the stability of evolution equation (1.1) with a local mixed-type damping.We only assume that the viscoelastic and frictional damping together cover a part of Ω; i.e., the support of a x b x ( ) ( ) + includes a sub-domain ω which satisfies the piecewise multiplier geometric condition (PMGC; Section 2 and Appendix), we do not make any additional assumptions on the location of viscoelastic damping; i.e., we do not assume that the viscoelastic-effect region includes a part of boundary, which is a necessity to obtain the decay estimate of the whole energy in almost all existing literature.We obtain the stability of problem (1.1) under such weak geometric condition (Theorems 3.1, 3.3, and 3.4), which reveals that the stability does not depend on the location of viscoelastic damping, but on whether the two types of damping in combination match the PMGC.
• In this article, we successfully establish a decay formula with almost smallest restrictions on the relaxation function g , which is only assumed to ensure the well-posedness of the problem (1.1) (Theorem 3.1).Moreover, after removing the fundamental conditions that g′ is controlled by g (like (1.4)), some optimal rates of uniform exponential/polynomial decay are obtained only under as small as conditions on the primitive G t ( ) of the relaxation function g t ( ) (Theorems 3.3 and 3.4).This implies that such weak condition on the primitive G t ( ) is enough to effect the asymptotic behavior and we can then obtain the optimal decay estimate.We do not need the much stronger condition on the derivative g t ( ) ′ , which is the key point to obtain the (optimal or uniform) stability in the past few decades.As you will see, our results break the persistent ideasome restrictions on the derivative g t ( ) ′ is necessary.And it is obvious that the restriction on relaxation function g here is weaker.Thus, we extend, improve, and cover the related stability results, even if for the entire damping case (or single viscoelastic damping system), the theorems here can also give stronger decay results.
• Thus, a distinct feature of the article is to study the uniform stability of (1.1) for local damping case under the PMGC (i.e., a x b x supp( ( ) ( )) + satisfying PMGC), which is a generalization of Γ-condition and a much weaker geometric restriction.An additional feature of our work is the generality of the relaxation function, where we only impose very few assumptions on the primitive G t ( ) of relaxation function g t ( ).In the framework of generality, a decay rate formula is constructed and some exactly optimal (so far best) decay rates are obtained by utilizing of the formula.In closing this part, we wish to point out that the purpose of this article is to minimize as much as possible the restrictions on a x b x , ( ) ( ), and g to study the stabilization for evolution equation with local mixed-type damping.To overcome these difficulties, we introduce some new ideas and techniques in this article, which are used to deal with the convolution energy term (more comments of this work are given in Sections 3 and 4).
Uniform decay estimates for the semi-linear wave equation  3 In conclusion, we discuss the stability problem of (1.1) for the local damping case.As you will see in Section 3, only under the local damping, for general relaxation function g and frictional function h, we give an important and useful prior estimate of the energy E t ( ) (Theorem 3.1).In particular, for some special g and h, we give some optimal decay rate (exponential and polynomial) only under local damping without any geometric restrictions on viscoelastic damping part (Theorem 3.3), these results are new even for the damping effecting on the entire domain (here notice that we don't assume the viscoelastic damping part include any boundary, which was considered in [33] for a special case).Moreover, if viscoelastic region include a part of boundary, we obtain the same decay properties for local damping and entire damping cases (Theorem 3.7).
The outline of the article is the following.Section 2 provides preliminaries on PMGC, assumptions, and well-posedness results.In Section 3, we present the main decay results concerning problem (1.1) and give some comments.We are devoted to deriving the complete proofs in Section 4. Finally, the article ends with Appendix A intended to assist the readers with some technical properties.Throughout the article, f f 1 2 and g g ≤ ≤ and g k g … represent some positive constants (possibly different from line to line).
Remark 2.2.One can see that multiplier geometric condition (or Γ-condition) is a special case (taking J 1 = in Definition 2.1) and their summed interior and boundary measures can be arbitrarily small.Moreover, in 1D case, the PMGC is satisfied by every nonempty subset of the interval Ω.But when space dimension n 2 ≥ , if ω Ω ⊂ satisfies the PMGC, then ω meas Ω 0 ( ) ∂ ∩ ∂ > .For more details and graphic examples, we refer the readers to [5] and figures there.
The following assumptions will be used in this article.
are nonnegative functions, and there exists a non-empty open subset ω Ω ⊂ satisfying PMGC such that where c 0 0 > is a constant.Moreover, if the space dimension n 1 = , we assume that +∞ is a non-increasing and locally absolutely continuous function with meas 0 0 J ( ) = and where c c , 0 1 2 > are positive constants.

Remark 2.3. (1)
The assumption I 1 ( ) implies the geometric control conditions on the domain of influence for the local mixed-type damping.From the assumption on the coefficients a x b x , ( ) ( ) in I 1 ( ), it can be seen that viscoelastic and frictional damping has effect only on a part ω of Ω, not in the entire domain.Furthermore, the effective regions of the viscoelastic damping may be arbitrary local sub-domains of Ω, not having to include a part of boundary of Ω.For the sake of clear and intuitive impression, we shall show the readers some graphic examples.For example, in Figure 1(a) and (b), the entire domain is a 2Ddisk, the shadow parts denote the damping regions where the viscoelastic and frictional effects work, and the viscoelastic damping effects in the dark grey sub-domains (i.e., a supp( ) includes the dark grey parts).In Figure 1(c), the entire system domain is a 2D-ring (light grey, grey, and dark grey parts), the grey and dark grey parts denote the viscoelastic-effect and frictional-effect region, and the viscoelastic damping effects in the dark grey sub-domains.The three cases in Figure 1 all meet I 1 ( ), but so far these three cases are still not considered in the previous literature (to the best of our knowledge) because they do not satisfy some specific conditions.Please see also more graphic examples and comments in Appendix A.
(2) The second distinct feature of this work is reflected in the restriction condition on relaxation function g in I 2 ( ).We drop the usual control conditions that g′ have to be controlled by g (like (1.4), [13, page 1990  (3) In I 3 ( ), we just impose the growth conditions on the frictional term h at infinity (which is necessary for uniform decay rates with frictional damping), while h is not required to satisfy any growth conditions at the origin (which is the critical region for stability).This condition has been used in [13,38].This time Uniform decay estimates for the semi-linear wave equation  5 we apply it into the issue of local mixed-type damping.Moreover, according to [11,13,38], it is possible to construct a convex, strictly increasing function Define the energy of a solution u of system (1.1) as Next, we state an existence and uniqueness result for system (1.1) for completeness.As for the proof, we refer the reader to [13,14].
Theorem 2.4.Suppose that (I 1 )-(I 3 ) hold.Then for any u H u L Ω , Ω 0 0 , the solution is a regular one satisfying Also, its energy satisfies that

Stability results and comments
In this section, we present our main stability theorems in several cases according to the conditions on the viscoelastic-effect region and the relaxation function.Moreover, we will also give some remarks and examples for better understanding of our theorems.If a x 0 ( ) ≡ in all Ω, no viscoelastic damping is present, i.e., model (1.2), which was studied in a great deal of work in the past several decades.Therefore, in this article, let us assume a x x max , Ω 0 { ( ) } ∈ > , i.e., viscoelastic damping is always present in (1.1).Further- more, we assume that b 0

Decay estimate for arbitrary local viscoelastic effects
Now, we shall give four decay results for the case of arbitrary local viscoelastic versus frictional dissipative effects.Please note that Theorem 3.1 is the fundamental theorem under almost minimal condition ensuring the well-posedness of problem (1.1).A uniform stability is shown in integral form in Theorem 3.1; one will obtain easily an explicit decay rates according to E t 0. ( ) ′ ≤ In the following three theorems, we give the explicitly quantified decay estimates by different restrictions on the primitive G t ( ) of the relaxation function g t ( ).It is worthy to note that we do not impose any assumption on the derivative g′ apart from the basic condition which ensures the well-posedness, while in the almost all previous literature, the control condition of g′ by g have to be imposed to obtain the decay results.On the other hand, one also has to assume much stronger geometric control conditions for the viscoelastic parts in the previous literature, while in this subsection, the geometric conditions are much weaker, not only we do not assume the viscoelastic parts include the boundary, but also we only restrict the total damping region satisfies the PMGC.These are all first considered in the stability of evolution equations involving viscoelastic effects.As will be seen, we here give optimal or so far best decay estimates under so weak conditions.Theorem 3.1.Suppose that (I 1 )-(I 3 ) hold.Then for every weak solution of problem (1.1), E t (1) Due to E t 0 ( ) ′ ≤ from (2.3), it is clear that Theorem 3.1 will give us a prior decay estimation for the energy E t ( ), In particular, when the frictional damping H is approximately linear (or no frictional damping is present), Theorem 3.1 tells us the energy E t ( ) is integrable, and then, it implies the energy decays at least by the rate t 1 1 ( ) + − .It is worth noting that both frictional and viscoelastic damping only exist in ω, which is a part of domain (not entire domain Ω) here, and the viscoelastic damping region is arbitrary subsets of Ω.We do not assume that viscoelastic-effect region reaches the boundary of Ω, while the later is necessary in the literature.
(2) We wish to point out that this estimate in Theorem 3.1 is very important for this problem.This is because in the previous results the estimate of viscoelastic energy is very difficult, and it is well known that the viscoelastic energy is only controlled by viscoelastic term.One has to make some additional assumptions in order to control it.For example, [13, page 1992, condition (15)], [37, Theorem 2.3, condition (2.10)] and [55, page 2134, condition (3.5)] had to be assumed in the previous works, while Theorem 3.1 ensures these assumptions hold automatically.Therefore, even if we drop these above mentioned conditions, the results in [13,37,55] still hold.(3) Our results obtained here are based on very weak condition for the relaxation function.Here, we only assume that it is non-increasing, and the additional restrictions that g s ( ) ′ is controlled by g s ( ) is dropped, while it is a necessary condition in order to obtain the decay rate of the energy in the almost all previous literature.Second, the problem here is a local mixed-type damping problem, the damping region only need to satisfy the PMGC condition.From the above-mentioned section and some related literature, the PMGC condition is very weak.In the case of disc domain, any small subregion including a diameter will satisfy the PMGC condition; that is, the entire damping region may only include any small part of the boundary, even its summed interior and boundary measures can be arbitrarily small.Moreover, if more conditions are imposed on the frictional term h, we then can give the explicitly optimal (so far best) decay rates.Now, we assume that c s hs c s s , 1 , where q 1 ≥ , and c 3 , c 4 are positive constants.
Uniform decay estimates for the semi-linear wave equation  7 Theorem 3.3.Assume that (I 1 )-(I 3 ), (3.1), and G s Then, the solution energy E t ( ) of the system (1.1) satisfies it follows that Theorem 3.4.Assume that (I 1 )-(I 3 ), (3.1), and (3.2) hold.Then, the solution energy E t ( ) of system (1.1) satisfies Here, we have set p In particular, if the function H is approximately linear in (2.2), and where + is a given strictly increasing and convex function with , and (I 1 )-(I 3 ), (3.3) hold.Then, the solution energy E t ( ) of the system (1.1) satisfies where s t ( ) satisfies the ordinary differential equation (ODE) and p H I 1 • In Theorem 3.3-Corollary 3.5, we only give very few restrictions on the primitive G t ( ) of the relaxation function g t ( ) and obtain some optimal (so far best) decay results.Note that here we do not assume that g′ is controlled by g, which is necessary in the literature to obtain the explicit decay rate.Therefore, here a breakthrough is that the asymptotic behavior can be dominated by G t ( ), not only by g′.From the knowledge of mathematical analysis, the restriction on G t ( ) is weaker than g′.Thus, as you can see, we break the used dominant view of the viscoelastic damping.In the previous literature, one always uses the term 3).Thus, additional condition on g′ has to be imposed.But these assumptions are just sufficient, not necessary.From Lemma 4.1, the viscoelastic energy can be controlled by the potential energy in integral form.Therefore, we can drop the traditional idea and then replace it by integral form (the assumption on the primitive G t ( )).
• Some examples of relaxation function: Then an immediate calculation deduces that , so such class of functions satisfies the condition in Theorems 3.3 and 3.4.But we see this class of relaxation functions do not meet the traditional conditions that g′ can be controlled by g.

The case: viscoelastic-effect region including a part of boundary
(3.5) implies the viscoelastic part includes a part of boundary, which tells us that the Poincarĺę inequality holds on the viscoelastic-effect region from the assumption (I 1 ).In this subsection, we shall study this case and establish a stability theorem.Then, we will compare our results for local mixed-damping case with the results in [13] for entire mixed-damping case.
In the existing results, for the mixed-type damping system, one focused on the decay problems in the following case: the total of viscoelastic and frictional damping covers the whole domain Ω, and the viscoelastic region must include a part of boundary Ω ∂ ; the relaxation function g can control its derivative g′.But if the total damping effects only in a local part, not covers the whole domain, the problem of the stability has not been studied until now.So in this subsection, we give some decay results for locally distributed mixed-type systems satisfying (3.5).Clearly, this case is a special situation of the above subsection.Thus, some estimates obtained in Section 3.1 can be used, then it will bring us great help to deduce the desired decay rates.On the other hand, it also implies that the result obtained in Theorem 3.1 is really useful and important for further research, and in fact, the result in Theorem 3.1 is a fundamental conclusion in stability problem of viscoelastic systems. Let where + is a given strictly increasing and convex function with where q 1 3 ≤ < .
where p H H I 3 Uniform decay estimates for the semi-linear wave equation  9 Remark 3.8.From Theorem 3.7, when (3.5) holds, the decay rates of the entire model will be dominated by the competing viscoelastic and frictional effects.This confirms that the decay results obtained in Theorem 3.7 are the same as that in [13].But we show that for local damping case under weaker conditions, it is the first time that the decay results here are obtained without the assumption [13, page 1992, (15)] for the viscoelastic wave equation.One of the distinct features is the results here are established under local damping condition, not entire damping.More importantly, we wish to point out here that another significant contribution is that we remove the assumption [13, page 1992, (15)], which automatically holds by Theorems 3.1-3.4,while in the previous literature, only several special cases were verified, such as [37,55].
In a word, in [13], the decay results must be based on the condition [13, page 1992, (15)] and the two types of damping effects in whole domain Ω.But here as you can see, we drop the condition [13, page 1992, (15)] and obtain the same decay rates under the local damping assumption; i.e., all of dampings only exist in a very limited part of Ω, which is due to the estimate obtained in Theorems 3.1 and 3.4.Therefore, Theorem 3.1 is essentially valuable, which will improve and extend a lot of related results.
Remark 3.9.In the present article, we study the stability of the mixed damping system with Dirichlet boundary conditions.Recently, the research on the stability of dynamic boundary damping systems is a frontier topic.Gao et al. [26,27] and Li et al. [40] studied the stability of wave equation with dynamic boundary conditions and obtained some decay results.For more research on dynamic boundary systems, see, e.g., [23,28,29] and references therein.However, viscoelastic damping is not involved in these studies.
In this article, some methods are introduced to deal with the viscoelastic damping term in system (1.1).We hope that these methods introduced here are also applicable to systems with dynamic boundary conditions.We will consider the related problems of the mixed damping systems with dynamic damping in the succedent work.

Proofs of main results
In this section, we prove our main stability theorems.Since we drop some important hypotheses here on relaxation function g x ( ) and the coefficients of damping terms a x ( ), b x ( ), which play a key role to obtain uniform decay rates in the previous literature, we have to look for some new techniques/approaches and reestablish several useful lemmas to prove our main results under such weaker conditions.

Lemmas and their proofs
In this subsection, first of all we shall establish several useful lemmas.
Let H be a real Hilbert space with scalar product , ⟨⋅ ⋅⟩ and norm ‖⋅‖.

Proof. A direct calculation shows that
. Then for any σ 1 < , we have According to the property of beta function, it is clear that f t t 1 σ σ Similarly, we can proof the other cases.□

Estimation of the total energy by its part on ω
The subset ω satisfies the PMGC.From Definition 2.1, we have, for some λ 0 > , Uniform decay estimates for the semi-linear wave equation  11 where Ω j and x j , j J 1, , = … denote the sets and the points given by the PMGC, respectively.Moreover, ) is the usual Euclidean distance to the subset of R n , and γ x x x x ν x Ω , 0 For m x x x j j ( ) = − , we define the C 1 vector field on Ω: Let ω be a subset of Ω satisfying the PMGC.Then, for n 1 ≥ and any S T 0 > ≥ , Here and δ 0, 1 is a constant which will be specified later; +∞ is a non-negative and non- increasing function with ψ 0 0 ( ) > , which will also be specified later.
The above two auxiliary functions M δ ( ) and K s δ ( ) were first introduced in [32, page 1512] and further used in [33,34].

Proof. Let us introduce the functional as
which will produce the negative terms According to the assumption I 1 ( ) and (1.1), we have, after integration by parts t u u g t sax us s x u g t s a x u s s u g t s a x u s s x where Thanks to the choice of ϒ and ψ j , only the boundary term on ∩∂ is nonvanishing in the first to third term of the right hand side of (4.3).But on this part of the boundary u 0 = , so that u 0 t = and u uν ν j j ∇ = ∂ .Therefore, the first to third term of the right hand side of (4.
Using ψ 1 j = on Q Ω \ j 1 and summing the resulting inequalities on j, we obtain Moreover,

s g t s a x u t u s s x b x h u u g t s a x u s s x n u g t s u t u s s x n b x h u u x
Therefore, for any n 1 ≥ , we have from (4.6) and (4.10), and (2.3), integrating (4.12) over T S , [ ] leads to (4.1).□

Estimation of the energy on ω
First of all, let us introduce three nonnegative functions a x a x , and

C ε ψ t a x u t x t C M δ ψ t K t s g t s a x u t u s s x t C ψ t b x h u t x t C ψ t G t u t x t
Proof.In view of (4.13),

ψ t b x u t x t ψ t u t B u t t ψ t A G B u t A G B B u t t ψ t u t A G B B u t t ψ t G t B u t A G B B u t t ψ t g t s B u t B u s s A G B B u t t ψ t b x h u A G B B u t t
Now, let us analysis the right hand side of (4.18) term by term.According to [33, Lemma 3.1], for the first term: for any ζ 0 1 > ,

ψ t u t A G B B u t t ψ t u t A G B B u t ψ t u t A G B B u t t ψ t u t A G B B u t t D ψ T E T ζ ψ t u x t C ζ ψ t b x u t x t
For the second term, from [33, page 7240-7241], where Similarly, we have for the third and fourth terms,
Analogously, by a similar argument used in the proof of (4.16), (4.17) holds.□ Lemma 4.5.For any ε 0 3 > , there exist constant C 4 and C ε 0 where t 0 is a positive number, being so large that , we obtain, using integration by parts and Young's inequality, for S T t  Ω 0 Uniform decay estimates for the semi-linear wave equation  21 Moreover, we observe from the assumptions on a x a x b x , , +∞ in mind, one obtains Similarly, we have For the last term on the right of (4.24), we infer that  Ω 0 Now, we have to analyze the first term on the right hand side of (4.24).For this purpose, we shall observe the following inequality: On the other hand, it is easy to deduce that Uniform decay estimates for the semi-linear wave equation  23 Then, we end the proof.
Proof.In view of (4.15), we have Take , In addition, use Lemma we infer that and ) in (4.31), we deduce that, for any S T t 0 Uniform decay estimates for the semi-linear wave equation  27 according to E t 0 ( ) ′ ≤ and the third item of Remark 2.3.On the other hand, a discussion similar to that in [44, page 1863] shows that Therefore, putting the above inequality into (4.37) and taking ε 0 sufficiently small yield that, for any S T t 0 Then, taking T t = * will lead to , it implies that Indeed, Theorem 3.3 is evident for q 1 = .What remains is to discuss the case for q 1 ≠ , which is divided into two cases.We start with the following case.
Case 1: q 1 3 < < .We break the proof of Case 1 into two steps.
and using Lemma 4.2, we have Thus, from (4.39), for any S T t 0 > ≥ > * , we obtain We prove (4.40) by induction.Clearly, it is true for j 0 = .Now, we assume it holds for j i 1 = − , i.e., Then, by Lemma 4.2, we obtain Thus, from (4.39), for any S T t 0 > ≥ > * , it holds that according to Lemma 4.2.Then, from (4.39), for any Step (ii) In this step, we show that there exists a positive constant r 1 such that By Lemma 4.2 and step (i), we infer that from (4.39), for any S T t 0 > ≥ > * , Uniform decay estimates for the semi-linear wave equation  29 which implies E t ( ) is integrable.Thus, we complete the proof for q 1 3 < < according to steps (i) and (ii).Analogously, using the same argument as Case 1, we can complete the proof of (4.41); here, we omit the details.Therefore, by virtue of (4.41), we have

E T G T s u s s x E T E T E T E T d d .
T ι Ω 0 2 1 1 1 q q q ι j q j 1 2   q q q p q p q q 1 2 1 2   q q q q q q q 1 2 1 2   Therefore, performing a standard argument using the above inequality can help us to complete the proof.Similar to Lemma 4.7, we have (for the sake of conciseness, we omit the details here), for S T t Then, using the standard argument as in [13, page 2007-2008] will lead us to finish the proof.□ So, it does not satisfy the conditions appearing in the previous literature (Appendix A for more examples).

−
depends explicitly on H 1 describing the viscoelastic damping, and t * is a positive constant.Remark 3.6.

−
depends explicitly on H and H 2 describing the viscoelastic and frictional damping. □

−
can be derived from (3.1).According to Theorem 3.1, we have

0 >
the same argument as in Lemmas 4.4 and 4.5, we have, for S T t 0 by Jin et al.The authors first introduced a pair of functions (4.2) (Lemma 4.3) and obtained the integrability of system energy (see [32, page 1526, (4.14)]) for any decreasing kernel.Since then, the auxiliary functions (4.2) have been widely used in the evolution systems with viscoelastic effects.
[45] the help of (4.2), Mustafa[45]obtained general decay rates for nonlinear viscoelastic equation: If we multiply the above equation by χ t ( ) and integrate from T to S, then we will complete the proof by Uniform decay estimates for the semi-linear wave equation  13 Together with the above inequality and (4.5), we obtain 11) Uniform decay estimates for the semi-linear wave equation  17 Lemma 2.2], [14, page 1315, (2.6)], for the sake of conciseness, we omit it here.
(boundary-related condition) does not appear in (I 1 ) here.Thus, we cannot use Poincaré's inequality directly, i.e., effect region in the previously related works.In this way, we can solve the arbitrary local viscoelastic effects case after removing the fundamental Boundary-related condition that the viscoelasticeffect region includes a part of the domain boundary.
4.8.We can reiterate the above argument and obtain a constant r 0 2 > such that Please note we cannot reach the decay rate t 1 in finite steps due to lack of more information of relaxation function.Once we obtained the specific information, the decay rate t 1 Then, using [4, Lemma 2.2], it follows that Uniform decay estimates for the semi-linear wave equation  31We finish the proof.The proof of Corollary 3.5.Suppose that H s s ( ).By Theorem 3.1, E t ( ) is integrable.Thus, by the assumption (3.3) and Jensen's inequality, we infer 4.5 Proof of Theorem 3.7Proof of Theorem 3.7.Note (3.5), we can use directly Poincaré's inequality for the term