Decay Rates for Solutions of a Timoshenko System with a Memory Condition at the Boundary

We consider a Timoshenko system with memory condition at the boundary and we study the asymptotic behavior of the corresponding solutions. We prove that the energy decay with the same rate of decay of the relaxation functions, that is, the energy decays exponentially when the relaxation functions decays exponentially and polynomially when the relaxation functions decays polynomially.


Introduction
The main purpose of this work is to study the asymptotic behavior of the solutions of a Timoshenko system with boundary conditions of memory type.To formalize this problem, take Ω an open bounded set of R n with smooth boundary Γ and assume that Γ can be divided into two parts (1.1) Denote by ν(x) the unit normal vector at x ∈ Γ outside of Ω and consider the following initial boundary value problem: ) u + t 0 g 1 (t − s) ∂u ∂ν (s)ds = 0 on Γ 1 × (0,∞), (1.5) v + t 0 g 2 (t − s) ∂v ∂ν (s)ds = 0 on Γ 1 × (0,∞), (1.6) u(0,x),v(0,x) = u 0 (x),v 0 (x) , u t (0,x),v t (0,x) = u 1 (x),v 1 (x) in Ω. (1.7) Here, u is the deflection of the beam from its equilibrium and v is the total rotatory angle of the beam at x, for those precise physical meaning, see Timoshenko [13].We will assume in the sequel that α is a sufficiently small positive number, β > nα, and the relaxation functions g i are positive and nondecreasing and the function f ∈ C 1 (R) satisfies f (s)s ≥ 0, ∀s ∈ R. (1.8) Additionally, we suppose that f is superlinear, that is, for some δ > 0 with the following growth conditions: for some c > 0 and ρ ≥ 1 such that (n − 2)ρ ≤ n.The integral equations (1.5) and (1.6) describe the memory effects which can be caused, for example, by the interaction with another viscoelastic element.Also, we will assume that there exists x 0 ∈ R n such that As an example of a set Ω satisfying those properties, we can consider the domain shown in Figure 1.1.

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Let m(x) = x − x 0 .Note that the compactness of Γ 1 implies that there exists a small positive constant δ 0 such that (1.12) Frictional dissipative boundary condition for the Timoshenko system was studied by several authors, see, for example, [4,6,11,12] among others.Concerning the memory condition at the boundary we can cite the following works: in [1], Ciarletta established theorems of existence, uniqueness, and asymptotic stability for a linear model of heat conduction.In this case the memory condition describes a boundary that can absorb heat and due to the hereditary term, can retain part of it.In [3], Fabrizio and Morro considered a linear electromagnetic model and proved the existence, uniqueness, and asymptotic stability of the solutions.In [7], Muñoz Rivera and Andrade showed exponential stability for a nonhomogeneous anisotropic system when the resolvent kernel of the memory is of exponential type.They used multiplier technics and a compactness argument.
Nonlinear one-dimensional wave equation with memory condition on the boundary was studied by Qin [9].He showed existence, uniqueness, and stability of global solutions provided the initial data is small in H 3 × H 2 .This result was improved by Muñoz Rivera and Andrade [8].They only supposed small initial data in H 2 × H 1 .See also de Lima Santos [2].
In this paper, we show that the solutions of the coupled system (1.2)-(1.7)decays uniformly in time with the same rate of decay of the relaxation functions.More precisely, denoting by k 1 and k 2 the resolvent kernels of −g 1 /g 1 (0) and −g 2 /g 2 (0), respectively, we show that the solution decays exponentially to zero provided k 1 and k 2 decays exponentially to zero.When the resolvent kernels k 1 and k 2 decays polynomially, we show that the corresponding solution also decays polynomially to zero.The method used is based on the construction of a suitable Lyapunov functional ᏸ satisfying for some positive constants c 1 ,c 2 ,γ, and α.Note that, because of condition (1.4) the solution of system (1.2)-(1.7)must belong to the following space: The notations we use in this paper are standard and can be found in Lions' book [5].In the sequel, by c (sometimes c 1 ,c 2 ,...) we denote various positive constants independent of t and on the initial data.The organization of this paper is as follows.In Section 2, we establish an existence and regularity result.In Section 3, we prove the uniform rate of exponential decay.Finally, in Section 4, we prove the uniform rate of polynomial decay.

Existence and regularity
In this section, we study the existence and regularity of solutions for the Timoshenko system (1. ( Applying the Volterra's inverse operator, we get where the resolvent kernels satisfy Denoting by τ 1 = 1/g 1 (0) and τ 2 = 1/g 2 (0) the normal derivatives of u and v can be written as (2.5) Reciprocally, taking initial data such that u 0 = v 0 = 0 on Γ 1 , identities (2.5) imply (1.5) and (1.6).Since we are interested in relaxation functions of exponential or polynomial type and identities (2.5) involve the resolvent kernels k i , we want to know if k i has the same properties.The following lemma answers this question.
Let h be a relaxation function and k its resolvent kernel, that is, ) (2.9) but this is contradictory.Therefore k(t) > 0 for all t ∈ R + 0 .Now, fix , such that 0 < < γ − c 0 and denote by k (t) := e t k(t), h (t) := e t h(t). (2.11) Multiplying (2.6) by e t we get (2.12) Therefore, which implies our first assertion.To show the second part consider the following notations: .17) which proves our second assertion.
Remark 2.2.The finiteness of the constant c p can be found in [10, Lemma 7.4].
Due to Lemma 2.1, in the remainder of this paper, we will use (2.5) instead of (1.5) and (1.6).Denote by The next lemma gives an identity for the convolution product.
The proof of this lemma follows by differentiating the term g2ϕ.
The well-posedness of system (1.2)-(1.7) is given by the following theorem. (2.20) then there exists only one strong solution (u,v) of the Timoshenko system (1.2)-

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This theorem can be proved using the standard Galerkin method, for this reason we omit it here.

Exponential decay
In this section, we study the asymptotic behavior of the solutions of system (1.2)-(1.7)when the resolvent kernels k 1 and k 2 are exponentially decreasing, that is, there exist positive constants b 1 and b 2 such that Note that these conditions imply that Our point of departure will be to establish some inequalities for the strong solution of Timoshenko system (1.2)-(1.7).For this end, we introduce the functional Summing the above identities, substituting the boundary terms by (2.5), and using Lemma 2.3 our conclusion follows.
Let θ > 0 be a small constant and define the following functional: The following lemma plays an important role for the construction of the Lyapunov functional.

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Performing an integration by parts, we get (3.10) Similarly, using (1.3) instead of (1.2) we get (3.11) Summing these two last inequalities, using Poincaré's inequality and taking θ small enough our conclusion follows.
We introduce the Lyapunov functional with N > 0. Using Young's inequality and taking N large enough we find that for some positive constants q 0 and q 1 .We will show later that the functional ᏸ satisfies the inequality of the following lemma.Proof.First, suppose that γ 0 < γ 1 .Define F(t) by Then (3.17) Integrating from 0 to t we arrive to Now, we will assume that γ 0 ≥ γ 1 , and we get Integrating from 0 to t, we obtain Since t ≤ (γ 1 − )e (γ1− )t for any 0 < < γ 1 we conclude that This completes the proof.
Finally, we will show the main result of this section.
If the resolvent kernels k 1 and k 2 satisfy (3.1), then there exist positive constants α 1 and γ 1 such that Proof.We will prove this result for strong solutions, that is, for solutions with initial data (u 0 ,v 0 ) ∈ (H 2 (Ω) ∩ V ) 2 and (u 1 ,v 1 ) ∈ V 2 satisfying the compatibility conditions (2.21).Our conclusion follows by standard density arguments.

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Using Lemmas 3.1 and 3.2, condition (1.12), and Young's inequality we get ∂u ∂ν ∂v ∂ν for any > 0. Choosing N large enough, fixing = δ 0 , and using the inequalities we arrive to where R(t) = k 1 (t) + k 2 (t) and q 2 > 0 is a small constant.Here we have used assumptions (3.1) in order to obtain the following estimates: for some boundary terms in (3.23).Finally, in view of (3.13) we conclude that From the exponential decay of k 1 , k 2 , and Lemma 3.3 there exist positive constants c and γ 1 such that From inequality (3.13) our conclusion follows.

Polynomial rate of decay
Here our attention will be focused on the uniform rate of decay when the resolvent kernels k 1 and k 2 decay polynomially like (1 + t) −p .In this case we will show that the solution also decays polynomially with the same rate.Therefore, we will assume that the resolvent kernels k 1 and k 2 satisfy for some p > 1 and some positive constants b 1 and b 2 .The following lemmas will play an important role in the sequel.
Theorem 4.3.Take (u 0 ,v 0 ) ∈ V 2 and (u 1 ,v 1 ) ∈ [L 2 (Ω)] 2 .If the resolvent kernels k 1 and k 2 satisfy conditions (4.1), then there exists a positive constant c such that Proof.We will prove this result for strong solutions, that is, for solutions with initial data (u 0 ,v 0 ) ∈ (H 2 (Ω) ∩ V ) 2 and (u 1 ,v 1 ) ∈ V 2 satisfying the compatibility conditions (2.21).Our conclusion will follow by standard density arguments.We define the functional ᏸ as in (3.12) therefore we have the equivalence relation given in (3.13) again.Combining Lemmas 3.1 and 3.2 we get 544 Decay rates for solutions of a Timoshenko system for some positive constants c 1 and c 2 .Using hypothesis (4.1) we obtain Denote by (4.10) Using the following estimates: inequality (4.9) can be written as Fix 0 < r < 1 such that 1/(p + 1) < r < p/(p + 1).Under this condition we have Using this estimate and Lemma 4.1 we get On the other hand, since the energy is bounded we have Substitution of (4.14) and (4.15) into (4.12)we arrive to Taking into account inequality (3.13) we conclude that Therefore, from Lemma 4.2 we conclude that

.19)
Under this condition applying Lemma 4.1 for r = 0 we get

.20)
Using these inequalities instead of (4.14) and reasoning in the same way as above, we conclude that

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: