A New General Decay Rate of Wave Equation with Memory-Type Boundary Control

Of interest is a wave equation with memory-type boundary oscillations, in which the forced oscillations of the rod is given by a memory term at the boundary. We establish a new general decay rate to the system. And it possesses the character of damped oscillations and tends to a finite value for a large time. By assuming the resolvent kernel that is more general than those in previous papers, we establish a more general energy decay result. Hence the result improves earlier results in the literature.


Introduction
It is well-known that if we add a damping to a system, the amplitude of the oscillations can be reduced very fast. e memory term can be as a damping (viscoelastic damping) which is weaker than frictional damping. For viscoelastic materials, Boltzmann theory gives us that the stress-strain viscoelastic law depending on a relaxation measure, see Prüss [1] and Eden et al. [2]. Based on the Boltzmann principle, the viscoelastic stress-strain relations can be generally given by a convolution term, which can be regarded as a lower order perturbation and can also be regarded as a kind of memory effect, for instance, g * u. And we call the function g(t) memory kernel. One can find a detail derivation on some systems with memory in [3].
To motivate our work, we start with some known results on wave equation with memory-type oscillations. e general wave equation with viscoelastic term in the internal feedback Messaoudi and Messaoudi [4,5] studied F(u) � 0 and F(u) � |u| ρ u, by introducing the assumption g ′ (t) ≤ − ξ(t)g(t), and obtained the energy decays exponentially (polynomially) as g decays exponentially (polynomially), respectively.
Lasiecka et al. [6] considered the general assumption on g: g ′ (t) ≤ − H(g(t)) to establish general decay of energy.
Here H, which was introduced by Alabau-Boussouira and Cannarsa [7], is strictly convex and increasing function. Cavalcanti et al. [8,9], Lasiecka and Wang [10], Mustafa and Messaoudi [11], and Xiao and Liang [12] also used this assumption to obtain some general decay rates of corresponding models. In recent papers [13][14][15], the authors investigated three classes of viscoelastic wave equation as in [4,5] and established optimal and explicit decay results of energy by adopting the assumption on g: g ′ (t) ≤ − ξ(t)H(g(t)).
In this paper, we considered the following wave equation with boundary oscillations of memory type: where Ω ⊂ R n (n ≥ 1) is a bounded domain with smooth boundary Γ, Γ � Γ 0 ∪ Γ 1 , and Γ 0 and Γ 1 are closed and disjoint with measure (Γ 0 ) > 0. ] is the unit outward normal to Γ. For wave equation with memory-type boundary oscillations, it can be regarded as a wave equation with viscoelastic damping at the boundary. Santos [16] considered a one-dimensional wave equation with memory conditions at the boundary, respectively. He proved that the energy of solutions decays exponentially (polynomially) as k and k ′ decay exponentially (polynomially). Here k is the resolvent kernel of (− g ′ /g(0)). Santos et al. [17] extended the results in [16] to an n-dimensional wave equation of Kirchhoff type with memory-type boundary. ey proved the global existence of solutions and obtained that the energy of solution decays uniformly with the same rate of decay k under the same conditions on k and k ′ , which improves the results in [18] by Park et al. Santos and Junior [19] obtained a similar result for plate equation with memory-type boundary. We also mention the work of Cavalcanti et al. [20], where the authors showed the global existence and the uniform decay of solutions to a semilinear wave equation with memorytype boundary condition and a nonlinear boundary source. Messaoudi and Soufyane [21] considered a general assumption on k ′ : k ″ ≥ − ξ(t)k ′ (t) and established a general decay result. Wu [22] used this assumption to study a wave Kirchhoff-type wave equation with a boundary control of memory type. For nonlinear wave equations with memorytype boundary condition, we refer to Cavalcanti and Guesmia [23], Feng [24], Feng et al. [25][26][27], Muñoz Rivera and Andrade [28], and Zhang [29].
Concerning the system (2), Mustafa [30], by assuming the function k: k ″ (t) ≥ H(− k ′ (t)), where k is the resolvent kernel of (− g ′ /g(0)), established a general decay of solutions of the form Here and D is a positive C 1 function with D(0) � 0, and H 0 is strictly increasing and strictly convex C 2 function on (0, r]. In particular, for H(t) � t p , i.e., k ″ ≥ c(− k ′ ) p , the author proved the energy decay holds for 1 ≤ p < (3/2). Whether can the range be extended to a more larger range? In this paper, we give a positive answer to study problem (2) and extend the result to get a more general decay rate. In particular, we obtain that the energy result holds for H(t) � t p with the full admissible range 1 ≤ p < 2. More exactly, by assuming the relaxation function k with minimal conditions on , where H is linear or strictly increasing and strictly convex functions of class C 2 (R + ), we establish an optimal explicit and general energy decay result. In particular, the energy result holds for H(t) � t p with the range p ∈ [1, 2) instead of p ∈ [1, (3/2)) in [30]. Hence our results extend and improve the stability results in [30] and also in [16][17][18]21]. We mainly adopt the idea of [14,15,31] and some properties of convex function developed in [7,32]. e remaining of the paper is organized as follows: in Section 2, we propose some preliminaries. In Section 3, main results are given. Section 4 is devoted to proving the general decay result.

Preliminaries
Taking the derivative of (2) with respect to t, we shall see that We denote the resolvent kernel of (− g ′ /g(0)) by k satisfying for t ≥ 0: Using Volterra's inverse operator and taking α � (1/g(0)), we have Assume u 0 � 0 on Γ 1 in this paper, we get In the following, we use boundary conditions (8) instead of (2).
As in [30], we consider the following assumption: (A1) ere exists a fixed point x 0 ∈ R 2 and some con- For the kernel k, we assume (A2) k: R + ⟶ R + is nonincreasing and twice differentiable function satisfying for any t ≥ 0, (A3) ere exist a C 1 function H: which is linear or is strictly increasing and strictly convex function of class where η(t) is C 1 nonincreasing continuous function.
is nonincreasing and nonnegative, we can get en for some t 1 ≥ 0 large, erefore we obtain that there exist two positive constants a and b such that for any t ∈ [0, is implies that there exists a constant d > 0 such that e proof is done.

Main Results
e well-posedness result is given in [30] proved by using the Faedo-Galerkin method as in [17].

and then problem (2) admits a unique solution u satisfying
e total energy of the system is defined by where We can get the following stability result.

Theorem 2. Assume k satisfies (A1)-(A3) and further
where and where c 1 , c 3 , and c 2 ≤ 1 are positive constants. (23), the energy result holds for H(t) � t p with the full admissible range p ∈ [1, 2) instead of p ∈ [1, (3/2)). If the viscoelastic term is as internal feedback, Lasiecka and Wang [10] provided the proof for optimal decay rates of second-order systems in the full admissible range [1,2).
At last, we show two examples to illustrate explicit formulas for the decay rates of the energy, which can be found in the studies of Mustafa and Mustafa [14,15].
we can deduce that the function H satisfies (A3) on (0, r] for any 0 < r < 1. en, By part 1 of (23), we get As c 2 ≤ 1, this is slower rate than [− k ′ (t)]. In addition, From part 2 of (23), we infer that for large t which is the same rate as [− k ′ (t)].

Proof of Main Result
To prove eorem 2, we need the following lemmas.

Lemma 2. Define the functional Φ(t) by
en we can get for any t ≥ t 1 , Proof. From the same arguments as in the study of Mustafa [30], we can obtain It follows from Young's inequality that for any ε > 0, By using Young's inequality, we obtain 4 Mathematical Problems in Engineering Hölder's inequality implies which, together with (37), gives us that Inserting (39) into (35), we obtain for any ε > 0, Noting that using lim t⟶∞ k(t) � 0 and taking ε > 0 small enough, we can get (33) from (34) and (40). e proof is done.

□
To get the optimal energy decay, we need the following estimate.

Lemma 3. e functional Ψ(t) is defined by
which satisfies for any t > 0, Proof. Differentiating Ψ(t) with respect to t, we get In view of Young's and Hölder's inequalities, we obtain Mathematical Problems in Engineering 5 en we can get (43) following from the fact (46) e proof is complete.
where N > 0 is a constant that will be taken later. Clearly we can take N a large value to get Recalling k ″ � δk ′ + h, combining (30) and (33), we conclude that for any t > t 1 , Noting − k ′ > 0 and k ″ > 0, for each s ∈ [0, ∞), we shall see below, It follows from Lebesgue dominated convergence theorem that erefore there exist 0 < c < 1 such that if δ < c, then And then we choose N a larger value that and take δ > 0 satisfying is implies en there exists a positive constant β such that for large (56) By (17) and (30), we get en from (56), we infer that there exists a constant χ > 0 such that , and using (58), we know that In the sequel, we consider two cases.

Case 1.
e particular case H(t) � t p .
(I) p � 1. Multiplying (59) by η(t), and using (19) and (A2)-(A3), we have Since η(t) is a nonincreasing continuous function and η ′ (t) ≤ 0 for a.e. t, then In view of ηF + cE ∼ E, we obtain that there exist two positive constants c 1 , c 2 > 0, Define G(t) by It follows from (43) and (56) that E(t) ≥ 0, and for any t ≥ t 1 , en there exists a certain constant β 1 > 0, is gives us Define we know that Without loss of generality assuming t 1 so large that I(t 1 ) > 0, then Using Jensen's inequality and by (30) and (A2)-(A3), we can derive from (56) that for some constant q > 0, We multiply (71) by E p− 1 (t) and use (19) to deduce By Young's inequality, we have for any ε 1 > 0, Taking en there exists a certain constant q 0 > 0 such that from which we obtain where c 3 is a positive constant. Combining (I) and (II) and using the boundedness of η(t) and E(t), we can get (23).
Case 2. e general case. Define In view of (67), we can take 0 < q < 1 such that Without loss of generality, we assume that I(t) > 0 for all t ≥ t 1 . On the other hand, we define From (30), we can easily get π(t) ≤ − cΕ ′ (t). As H(t) is strictly convex on (0, r] and H(0) � 0, we see that

Data Availability
No data were used during this study.

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.