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

Abstract

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.

1. Introduction

It is well-known that if we add a damping to a system, the amplitude of the oscillations can be reduced very fast. The 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, . And we call the function 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. The general wave equation with viscoelastic term in the internal feedback

Messaoudi and Messaoudi [4, 5] studied and , by introducing the assumption , and obtained the energy decays exponentially (polynomially) as decays exponentially (polynomially), respectively.

Lasiecka et al. [6] considered the general assumption on : to establish general decay of energy. Here , 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 [13–15], 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 : .

In this paper, we considered the following wave equation with boundary oscillations of memory type:where is a bounded domain with smooth boundary , , and and are closed and disjoint with measure . is the unit outward normal to .

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 and decay exponentially (polynomially). Here is the resolvent kernel of . Santos et al. [17] extended the results in [16] to an n-dimensional wave equation of Kirchhoff type with memory-type boundary. They proved the global existence of solutions and obtained that the energy of solution decays uniformly with the same rate of decay under the same conditions on and , 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 memory-type boundary condition and a nonlinear boundary source. Messaoudi and Soufyane [21] considered a general assumption on : 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 memory-type boundary condition, we refer to Cavalcanti and Guesmia [23], Feng [24], Feng et al. [25–27], Muñoz Rivera and Andrade [28], and Zhang [29].

Concerning the system (2), Mustafa [30], by assuming the function : , where is the resolvent kernel of , established a general decay of solutions of the form

Hereand is a positive function with , and is strictly increasing and strictly convex function on . In particular, for , i.e., , the author proved the energy decay holds for . 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 with the full admissible range . More exactly, by assuming the relaxation function with minimal conditions on , i.e., , where is linear or strictly increasing and strictly convex functions of class , we establish an optimal explicit and general energy decay result. In particular, the energy result holds for with the range instead of in [30]. Hence our results extend and improve the stability results in [30] and also in [16–18, 21]. We mainly adopt the idea of [14, 15, 31] and some properties of convex function developed in [7, 32].

The 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.

2. Preliminaries

Taking the derivative of (2) with respect to , we shall see that

We denote the resolvent kernel of by satisfying for :

Using Volterra’s inverse operator and taking , we have

Assume on 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)There exists a fixed point and some constant such that for , For the kernel , we assume(A2) is nonincreasing and twice differentiable function satisfying for any ,(A3)There exist a function , with , which is linear or is strictly increasing and strictly convex function of class on , such that where is nonincreasing continuous function.

Remark 2.1. If assuming further since and is nonincreasing and nonnegative, we can getThen for some large,Noting that is nonincreasing, , and , we have for any , and for any ,Therefore we obtain that there exist two positive constants and such that for any ,Then for any ,This implies that there exists a constant such that for any ,The proof is done.

3. Main Results

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

Theorem 1. Assume that (A1) and (A2) hold. Let , and then problem (2) admits a unique solution satisfyingwhere .

The total energy of the system is defined bywhere

We can get the following stability result.

Theorem 2. Assume satisfies (A1)–(A3) and further Then there exist such thatwhereand . In particular, if , then for any ,where , and are positive constants.

Remark 3.1. From (23), the energy result holds for with the full admissible range instead of . 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 .

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].

Example 1. Take with , we get , where . Sincewe can deduce that the function satisfies (A3) on for any . Then,

Example 2. Consider with , we get . Clearly,By part 1 of (23), we getAs , this is slower rate than . In addition,From part 2 of (23), we infer that for large which is the same rate as .

4. Proof of Main Result

To prove Theorem 2, we need the following lemmas.

4.1. Technical Lemmas

Lemma 1. The total energy functional satisfies for any ,

Proof. See [30].

As in [31], for , we introduce

Lemma 2. Define the functional by

Then we can get for any ,

Proof. From the same arguments as in the study of Mustafa [30], we can obtainIt follows from Young’s inequality that for any ,Recalling , where ; then we have from (8),By using Young’s inequality, we obtainHölder’s inequality implieswhich, together with (37), gives us thatInserting (39) into (35), we obtain for any ,Noting thatusing and taking small enough, we can get (33) from (34) and (40). The proof is done.

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

Lemma 3. The functional is defined bywhich satisfies for any ,

Proof. Differentiating with respect to , we getIn view of Young’s and Hölder’s inequalities, we obtainThen we can get (43) following from the factThe proof is complete.

4.2. Proof of Theorem 2

Proof. Define the functional bywhere is a constant that will be taken later. Clearly we can take a large value to getRecalling , combining (30) and (33), we conclude that for any ,Noting and , for each , we shall see below,It follows from Lebesgue dominated convergence theorem thatTherefore there exist such that if , thenAnd then we choose a larger value thatand take satisfyingThis impliesThen there exists a positive constant such that for large ,By (17) and (30), we getThen from (56), we infer that there exists a constant such thatDenoting , and using (58), we know that

In the sequel, we consider two cases.

Case 1. The particular case . (I) . Multiplying (59) by , and using (19) and (A2)-(A3), we have Since is a nonincreasing continuous function and for a.e. , then In view of , we obtain that there exist two positive constants , (II) . Define by It follows from (43) and (56) that , and for any , Then there exists a certain constant , This gives us Hence Define we know that Without loss of generality assuming so large that , then Using Jensen’s inequality and by (30) and (A2)-(A3), we can derive from (56) that for some constant , We multiply (71) by and use (19) to deduce By Young’s inequality, we have for any , Taking , we conclude Define . Multiplying (74) by , we have Then there exists a certain constant such that from which we obtain where is a positive constant. Combining (I) and (II) and using the boundedness of and , we can get (23).

Case 2. The general case.
DefineIn view of (67), we can take such thatWithout loss of generality, we assume that for all . On the other hand, we defineFrom (30), we can easily get . As is strictly convex on and , we see thatIt follows from Jensen’s inequality and (11) and (79) thatwhere , which is strictly convex and increasing function on of class , is called the extension of . We infer from (82) thatThen we can get from (59) that for any ,DenoteFor , we define bySince , , and , we get from (84) thatWe denote by the conjugate function of the convex function (see Arnold [33]), and thensatisfies Young’s inequality,Taking and , and using and (87), we haveWe multiply (90) by to arrive atThe functional is defined byThen we can easily obtain that there exist constants and such thatChoosing a suitable , and defining , from (91), we infer that for a constant ,It follows from that for any ,Using (93), we haveSince , then, noting the strict convexity of on , we know on . By (94), we conclude that there exists such that for any ,Integrating (97) over , we see thatDefineIt is to verify that is strictly decreasing on and . It follows thatCombining (96) and (100), we can obtain (21). This finishes the proof of Theorem 2

Data Availability

No data were used during this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (no. JBK1809025).