General Decay for the Degenerate Equation with a Memory Condition at the Boundary
We consider a degenerate equation with a memory condition at the boundary. For a wider class of relaxation functions, we establish a more general decay result, from which the usual exponential and polynomial decay rates are only special cases.
The main purpose of this paper is to investigate the asymptotic behavior of the solutions of the degenerate equation with a memory condition at the boundary where is a bounded domain of with a smooth boundary and let us assume that , can be divided into two nonnull parts and and and for all which satisfies some appropriate conditions. is the unit outward normal to and the corresponding unit tangent vector. Here, the relaxation functions () are positive and nondecreasing, the function and and the constant , , represents Poisson’s ratio.
From the physical point of view, we know that the memory effect described in integral equations (3) and (4) can be caused by the interaction with another viscoelastic element. In fact, the boundary conditions (3) and (4) mean that is composed of a material which is clamped in a rigid body in and is clamped in a body with viscoelastic properties in the complementary part of its boundary named . Problems related to (1)–(5) are interesting not only from the point of view of PDE general theory, but also due to its applications in mechanics.
The existence of global solutions and exponential decay to the degenerate equation with has been investigated by several authors. See Cavalcanti et al.  and Menezes et al. . For instance, when is equal to 1, (1) describes the transverse deflection of beams. There exists a large body of literature regarding viscoelastic problems with the memory term acting in the domain or at the boundary (see [3–17]). Santos et al.  studied the asymptotic behavior of the solutions of a nonlinear wave equation of Kirchhoff type with boundary condition of memory type. Cavalcanti et al.  proved the uniform decay rates of solutions to a degenerate system with a memory condition at the boundary. Santos and Junior  investigated the stability of solutions for Kirchhoff plate equations with a boundary memory condition. Rivera et al.  showed the asymptotic behavior to a von Karman plate with boundary memory conditions. Park and Kang  studied the exponential decay for the Kirchhoff plate equations with nonlinear dissipation and boundary memory condition. They proved that the energy decays uniformly exponentially or algebraically with the same rate of decay as the relaxation functions. In the present work, we generalize the earlier decay results of the solution of (1)–(5). More precisely, we show that the energy decays at the rate similar to the relaxation functions, which are not necessarily decaying like polynomial or exponential functions. In fact, our result allows a larger class of relaxation functions. Recently, Messaoudi and Soufyane , Mustafa and Messaoudi , and Santos and Soufyane  proved the general decay for the wave equation, Timoshenko system, and von Karman plate system with viscoelastic boundary conditions, respectively.
The organization of this paper is as follows. In Section 2, we present some notations and material needed for our work and state the existence result to system (1)–(5). In Section 3, we prove the general decay of the solutions to the degenerate equation with a memory condition at the boundary.
Note that, because of condition (2), the solution of system (1)–(5) must belong to the following space: Let us define the bilinear form as follows: Since , we know that is equivalent to the norm on ; that is, and here and in the sequel, we denote by and generic positive constants. Simple calculation, based on the integration by parts formula, yields We assume that there exists such that If we denote the compactness of by , condition (12) implies that there exists a small positive constant such that , for all .
Next, we will use (3) and (4) to estimate the values and on . Denoting by the convolution product operator and differentiating (3) and (4), we arrive at the following Volterra equations: Applying the Volterra inverse operator, we get where the resolvent kernels satisfy Denoting that and , we have Therefore, we use (17) instead of the boundary conditions (3) and (4).
Let us denote that The following lemma states an important property of the convolution operator.
Lemma 1. For , one has
The proof of this lemma follows by differentiating the term .
Lemma 2 (see ). Suppose that , and ; then, any solution of satisfies and also
We formulate the following assumptions.(A1) Let satisfy Additionally, we suppose that is superlinear; that is, for some with the following growth condition: for some and such that .(A2); with , for all , and satisfy the following condition
3. General Decay
In this section, we show that the solution of system (1)–(5) may have a general decay not necessarily of exponential or polynomial type. For this we consider that the resolvent kernels satisfy the following hypothesis.
(H) is twice differentiable function such that and there exists a nonincreasing continuous function satisfying
The following identity will be used later.
Lemma 4 (see ). For every and for every , one has
Let us consider the following binary operator: Then applying the Holder inequality for we have Let us define the functional The following lemma plays an important role in the construction of the desired functional.
Proof. Differentiating using (1) and Lemma 4, we get Let us next examine the integrals over in (39). Since on , we have on and since Therefore, from (39) and (40), we have Using the Young inequality, we get where is a positive constant. Since the bilinear form is strictly coercive on , using the trace theory, we obtain where is a constant depending on , , and . Substituting inequalities (43)–(45) into (42) and taking into account that on , as well as (23) and (25), we have Since the boundary conditions (17) can be written as our conclusion follows.
Let us introduce the Lyapunov functional with . Now, we are in a position to show the main result of this paper.
Theorem 7. Let . Suppose that the resolvent kernels , satisfy the condition (H). Then, there exist constants such that, for some large enough, the solution of (1)–(5) satisfies Otherwise, for all , where
Proof. Applying inequality (36) with in Lemma 6 and from Lemma 5, we obtain
We take and so small such that
Since and then choosing large enough, we obtain
On the other hand, we can choose even larger so that
If , , then, using (30) and (33), we have
Using the fact that is a nonincreasing continuous function as and are nonincreasing, and so is differentiable, with , for a.e. , then we infer that
Since using (55),
we obtain, for some positive constant ,
Case 1. If on , then (60) reduces to A simple integration over yields By using (33) and (59), we then obtain for some positive constant Thus, estimate (49) is proved.
Case 2. If on , then (60) gives where In this case, we introduce A simple differentiation of , using (64), leads to Again, a simple integration over yields which implies, for all , By using (59), we deduce that Consequently, by the boundedness of , (50) is established.
Remark 8. Note that the exponential and polynomial decay estimates are only particular cases of (49) and (50). More precisely, we have exponential decay for and and polynomial decay for and , where and are positive constants.
Example 9. As in , we give some examples to illustrate the energy decay rates given by (49).(1)If , , then, for , where . For suitably chosen positive constants and , satisfies (H) and (49) gives (2)If , , and , , then, for , where . Then The aforementioned two examples are included in the following more general one.(3)For any nonincreasing functions , , which satisfy (H), are also nonincreasing differentiable functions, and , for some , and (49) gives
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2012R1A1A3011630).
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