# Optimal Polynomial Decay to Coupled Wave Equations and Its Numerical Properties

Authors: R. F. C. Lobato; S. M. S. Cordeiro; M. L. Santos; D. S. Almeida Júnior
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

In this work we consider a coupled system of two weakly dissipative wave equations. We show
that the solution of this system decays polynomially and the decay rate is optimal. Computational experiments are conducted in the one-dimensional case in order to show that the energies properties are preserved. In particular, we use finite differences and also spectral methods.


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## Body

## 1. Introduction

This paper is concerned with the polynomial stability and the optimal rate of decay of theC0-semigroup of the coupled system of two wave equations:
(1)utt-Δu+ut+αv=0inΩ×]0,∞[,(2)vtt-Δv+αu=0inΩ×]0,∞[,(3)u=v=0on⁡Γ×]0,∞[,(4)(u(x,0),v(x,0))=(u0,v0),(ut(x,0),vt(x,0))=(u1(x),v1(x)),inΩ,
where Ω is an open bounded set of Rn with smooth boundary Γ. Here α is a small real positive constant.This problem is motivated by an analogous problem in ordinary differential equations for coupled oscillators and has potential application in isolation of objects from outside disturbances. As an example in engineering, rubber-like materials are used to absorb vibration or shield structures from vibration. Modeling of structures such as beam or plates sandwiched with rubber or similar materials will lead to equations similar to those (1)-(2).The exponential stability of coupled system of wave equations has been studied by Najafi et al. [1] in the case of two linear boundary feedback and by Komornik Rao [2] in the case of two nonlinear boundary feedback.We should mention that this initial boundary value problem (1)–(4) has been studied by many authors. For example, in [3] Alabau Boussoira proved that the energy of associated coupled weakly dissipative system decays polynomially as t-1 and this decay rate can improve since the initial data are more regular. In [4] Alabau Boussoira et al. showed the decay rates for an abstract coupled weakly dissipative system of using the same ideas introduced in [3]. For the case of the system (1)–(4) we can also cite the work of Santos et al. [5] where the authors proved t-1 polynomial decay and this decay rate can improve since the initial data are more regular. The question we are interested in is to study optimal polynomial decay rates to the system (1)–(4). That is, we prove that the associated semigroup decays at a rate t-1/2 and it is optimal. This work improves the results in [3–5]. Our result on the polynomial stability is based on Theorem 2.4 in [6].Naturally, the system (1)–(4) is dissipative. Indeed, the energy of the solutions defined by
(5)E(t):=12∫Ωut2dx+12∫Ω|∇u|2dx+12∫Ωvt2dx+12∫Ω|∇v|2dx+α∫Ωuvdx
is decreasing in t∈(0,∞), since
(6)ddtE(t)=-∫Ωut2dx≤0.To verify the asymptotic behavior of systems like (1)–(4), only the work of Najafi [7] treated numerically the problem of lack of exponential decay to a system of two coupled wave equations. Here, we use two numerical approaches: one is in finite differences and in the other one we use the spectral methods. In particular, we explore the properties of numerical energy associated with a particular discretization in finite differences.We conclude this introduction with an outline of the paper. In Section2, we show the existence of solutions for the system (1)–(4). In the Section 3 we showed that the system (1)–(4) is polynomially stable with optimal decay rate. Section 4 is dedicated with the numerical aspects. We derive a numerical energy and we show that it preserves the conservation and dissipation laws. We use also spectral methods to see the behavior of the numerical eigenvalues when the damping is weak. Finally, in Section 5, we finished this work with a conclusion.


## 2. The Semigroup Setting

In this section we will study the existence and uniqueness of strong and global solutions for the system (1)–(4) using the semigroup techniques.Let us denote byH=H01(Ω)×L2(Ω)×H01(Ω)×L2(Ω) the Hilbert space with internal product given by
(7)〈U,V〉H=∫Ω[∇u1·∇v1+u2v2+∇u3·∇v3+u4v4+α(u1v3+u3v1)]dx,
where U=(u1,u2,u3,u4)T and V=(v1,v2,v3,v4)T.Let us consider the unbounded operatorA in the energy space H with
(8)D(A)=H2(Ω)∩H01(Ω)×H01(Ω)×H2(Ω)∩H01(Ω)×H01(Ω),
defined by
(9)A=[0I00Δ-I-αI0000I-αI0Δ0].Puttingφ=ut and ψ=vt, (1) and (2) can be written as the following initial value problem:
(10)dUdt=AU,U(0)=U0,
with U=(u,φ,v,ψ)T and U0=(u0,u1,v0,v1)T. Using the internal product (7), we obtain
(11)Re〈AU,U〉H=-∫Ω‍|φ|2dx≤0.
Thus, A is a dissipative operator and therefore, using the Lumer-Phillips Theorem (see [8], Theorem 4.3), the operator A generates a C0-semigroup of contractions S(t)=eAt on H.Thus, we have the following result.Theorem 1.
Let one assume thatU0∈D(A); then there exists a unique solution U=(u,φ,v,ψ) to system (1)–(4) satisfying
(12)U∈C(R+;D(A))∩C1(R+;H).


## 3. Polynomial Decay and Optimal Result

To study the asymptotic behavior of the semigroup associated with (1)–(4), let us consider the spectral problem:
(13)-Δwν=λνwνinΩ,wν=0on⁡Γ,
where
(14)lim⁡ν→∞⁡λν=+∞.Theorem 2.
LetS(t) be C0-semigroup of contractions generated by A. Then it follows that S(t) is not exponentially stable.Proof.
Our main tool is to use Prüss’ result [9], which states that a semigroup eAt is exponentially stable if and only if the following conditions hold:
(15)iR⊂ρ(A)(resolventset),(16)∃K>0,∀U∈D(A),∀λ∈R:∥(iλI-A)-1∥H≤K.
This way, let us consider the resolvent equation
(17)(iλI-A)U=F,withλ∈R,F∈H;
that is,
(18)iλu-φ=f1,(19)iλφ-Δu+αv+φ=f2,(20)iλv-ψ=f3,(21)iλψ-Δv+αu=f4.
Let us takef1=f3=0 and f2=f4=wν. We look for solution of the form u=awν, v=bwν, φ=cwν, and ψ=dwν, with a,b,c,d∈C. From (19)–(21), we get a and b satisfying
(22)-λ2(a+b)wν+(a+b)λνwν+α(a+b)wν+cwν=2wν.
Now, choosingλ=λν+α, and using the above equation, we obtain c=2, so we have
(23)a=-2iλν+α,b=2iλν+α+1α,d=-2α+iλν+αα.
Therefore, we have(24)u=-2iλν+αwν,v=(2iλν+α+1α)wν,φ=2wν,ψ=-2α+iλν+ααwν.
Now we claim that(25)∥U∥H⟶+∞,asv⟶+∞.
In fact, using (7) and (24) and noting that
(26)lim⁡ν→∞⁡λν=∞,
we conclude that
(27)∥U∥H2≥∫Ω‍|(-2α+iλν+ααwν)|2dx⟶∞.
Recalling that(28)iλU-AU=F⟺U=(iλI-A)-1F,
it follows from (27) and (16) that S(t) is not exponentially stable.In order to show the polynomial decay from semigroup associated with the system (1)–(4), first let us consider the product in H of U=(u,φ,v,ψ)T∈D(A) with the resolvent equation of A; that is,
(29)iλ∥U∥H2-(AU,U)H=(F,U)H.
Now, noting that
(30)∫Ω‍|φ|2dx=-(AU,U)H,
we get
(31)iλ∥U∥H2+∫Ω|φ|2dx=(F,U)H.
Then taking the real part, we obtain
(32)∫Ω‍|φ|2dx≤∥U∥H∥F∥H.Lemma 3.
The strong solution of the system (1)–(4) given by Theorem 1 satisfies
(33)∫Ω‍|∇u|2dx+∫Ω‍|∇v|2dx+α∫Ω‍(uv¯+vu¯)dx≤∫Ω‍|ψ|2dx+K∥U∥H∥F∥H,
where K is a positive constant.Proof.
Multiplying equalities (19) and (21) by u¯ and v¯, respectively, integrating on Ω, and summing up the result we get
(34)iλ∫Ω‍φu¯dx︸:=I1+∫Ω|∇u|2dx+α∫Ωvu¯dx+iλ∫Ω‍ψv¯dx︸:=I2+∫Ω|∇v|2dx+α∫Ω∫uv¯dx=∫Ωf2u¯dx+∫Ωf4v¯dx.
Substitutingiλu¯ given in (18) into I1 and iλv¯ given in (20) into I2, we have
(35)∫Ω|∇u|2dx+∫Ω|∇v|2dx+α∫Ω(vu¯+uv¯)dx=∫Ω|φ|2dx+∫Ω|φ|2dx+∫Ωφf1¯dx+∫Ωψf3¯dx+∫Ωf2u¯dx+∫Ωf4v¯dx.
Using (32) our conclusion follows.Lemma 4.
The strong solution of the system (1)–(4) given by Theorem 1 satisfies
(36)(1-cλ)∫Ω‍|ψ|2dx≤c|λ|(∫Ω‍|∇v|2dx+∫Ω‍|∇u|2dx)+K|λ|2∥U∥H∥F∥H,
where K is a positive constant and |λ|>1 large enough.Proof.
Multiplying equality (21) by ψ¯ and integrating on Ω, we get
(37)iλ∫Ω‍|ψ|2dx=∫Ω‍Δvψ¯dx-α∫Ω‍uψ¯dx+∫Ω‍f4ψ¯dx.
Using equality (20) and Poincaré and Young inequalities, we have
(38)∫Ω‍|ψ|2dx≤c|λ|(∫Ω|∇v|2dx+∫Ω|∇u|2dx)+c|λ|∫Ω‍|ψ|2dx+K∥U∥H∥F∥H,
where c is a positive constant and |λ|>1 is large enough. The proof is now complete.Now, we are in the position to prove the main result of this paper.Theorem 5.
The semigroup associated with the system (1)–(4) is polynomially stable and
(39)∥S(t)U0∥H≤Kt∥U0∥D(A).
Moreover, this result is optimal.Proof.
From Lemmas3 and 4 and for |λ|>1, we have
(40)∥U∥H≤K|λ|2∥F∥H,
which is equivalent to
(41)∥(λI-A)-1∥≤K|λ|2.
Then using Theorem 2.4 in [6], we obtain
(42)∥S(t)A-1∥=O(t-1/2)⟹∥S(t)A-1F∥H≤Kt∥F∥H.
Since0∈ρ(A), it follows that A is onto over H; then taking AU0=F, we get
(43)∥S(t)U0∥H≤Kt∥U0∥D(A).
Therefore the solution decays polynomially.
To prove that the rate of decay is optimal, we will argue by contradiction. Suppose that the ratet-1/2 can be improved; for example, the rate is t-1/(2-ϵ) for some 0<ϵ<2. From Theorem 5.3 in [10], the operator
(44)|λ|-2+(ϵ/2)∥(λI-A)-1∥L(H)
should be limited, but this does not happen. For this, let us suppose that there exists a sequence (λμ)⊂R with lim⁡μ→∞|λμ|=∞ and (Uμ)⊂D(A) for (Fμ)⊂H such that
(45)(iλμI-A)Uμ=Fμ
is bounded in H and
(46)lim⁡μ→∞⁡|λμ|-2+(ϵ/2)∥Uμ∥H=∞.
Then, we can consider, for each μ∈N, Fμ=(0,wμ,0,wμ)T and Uμ=(uμ,φμ,vμ,ψμ)T, where, due to the boundary conditions, Uμ are in the form uμ=awμ, vμ=bwμ, φμ=cwμ, and ψμ=dwμ.
Then following the same steps as in the proof of Theorem2 we can conclude that
(47)|λμ|-2+(ϵ/2)∥Uμ∥H≥O(μϵ/2)⟶∞,asμ⟶∞.
Therefore the rate cannot be improved. The proof is now complete.


## 4. Numerical Approaches

In this section, we give a number of numerical experiments to certify the analytical results (see Figure15). We use two numerical approaches to this aim: the spectral method to compute the numerical eigenvalues and the explicit time integration method to compute the numerical solutions and to derive the associated numerical energy. In particular, the numerical energy preserves the conservation/dissipation of energy. Moreover, by means of numerical simulations we can note a lack of numerical exponential decay considering only damping (see Figure 17).Our numerical simulations are realized in the one-dimensional case. Then we consider the following one-dimensional system:(48)utt-uxx+αv+ut=0,in(0,L)×(0,T),(49)vtt-vxx+αu=0,in(0,L)×(0,T),(50)u(0,t)=u(L,t)=0,v(0,t)=v(L,t)=0,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkk0<t<T,(51)u(·,0)=u0(·),ut(·,0)=u1(·),v(·,0)=v0(·),vt(·,0)=v1(·).kkkkkkkkkkkkkkkkkkkk∀x∈(0,L).


### 4.1. Explicit Time Integration Method

For our purpose we define a computational mesh definingΔx=L/(J+1) and Δt=T/(N+1) for J,N∈N and the nets given by
(52)x0=0<x1=Δx<⋯<xJ=JΔx<xJ+1=L,t0=0<t1=Δt<⋯<tN=NΔt<tN+1=T,
where xj=jΔx and tn=nΔt for j=0,1,2,…,J+1 and n=0,1,2,…,N+1.We consider the following finite-difference discretization of (48)-(49):
(53)ujn+1-2ujn+ujn-1Δt2-uj+1n-2ujn+uj-1nΔx2+ujn+1-ujn-12Δt+αvjn=0,(54)vjn+1-2vjn+vjn-1Δt2-vj+1n-2vjn+vj-1nΔx2+αujn=0,
for all j=1,2,…,J, n=1,2,…,N. Such scheme is in fact explicit and its implementation is straightforward. The boundary conditions are
(55)u0n=uJ+1n=0,v0n=vJ+1n=0,∀n=1,2,…,N,
and the initial conditions are given by
(56)uj0=u(xj,0),uj1=uj0+Δtut(xj,0),∀j=1,2,…,J,(57)vj0=v(xj,0),vj1=vj0+Δtvt(xj,0),∀j=1,2,…,J.The discretizations above are consistent with truncation error of second order inΔx and Δt. Therefore, it is immediate by Lax Lemma [11] that (53)-(54) converges. For convenience, we use the stability criterion (taking into account the conservative case) due to Courant-Friedrichs-Lewy; namely, Δt≤Δx.


### 4.2. Energy Conserving Scheme

In this section we will build the numerical energy to system (53)–(57). Following the techniques performed, for example, in [12, 13], this energy is given by
(58)En:=Δx2∑j=0J‍[(ujn+1-ujnΔt)2+(uj+1n+1-ujn+1Δx)(uj+1n-ujnΔx)+(vjn+1-vjnΔt)2+(vj+1n+1-vjn+1Δx)(vj+1n-vjnΔx)+α(vjnujn+1+ujnvjn+1)(ujn+1-ujnΔt)2].We note thatEn is the discrete version of the continuous energy (5) in one-dimensional setting. Moreover, one can show that En decreases. Instead of computing the time derivative of the energy we can use the summation by parts. The discrete energy En is an important numerical instrument to certify our analytical results concerning the polynomial stabilization of system (53)–(57).We have the following result.Theorem 6 (discrete energy).
Let(ujn,vjn) be a solution of the finite difference scheme (53)–(57). Then for all Δt and Δx, the discrete rate of change of energy of the numerical scheme (53)–(57) at the tn instant of time is given by
(59)En-En-1Δt=-∑j=1J‍(ujn+1-ujn-12Δt)2,
for all n=1,…,N,N+1.Proof.
We shortened the proof. It is identical to the proof of Proposition 1 in [12]. Then, we can write
(60)En+Δt∑j=1J‍(ujn+1-ujn-12Δt)2=En-1,
from where we have
(61)En≤E0,∀n=1,…,N,N+1.
Considering the conservative case we get(62)En=E0,∀n=1,2,…,N.


### 4.3. Discussion on Numerical Results

In this section, we consider two numerical approaches to certify the analytical results established in this work: the pseudospectral method to compute numerically the eigenvalues and also using the numerical energyEn in (58).For spectral analysis we apply the spectral method according to Trefethen’s book [14]. Using this numerical technique, we can see the numerical behavior of eigenvalues of (48)-(49). To see how this system is affected by a lack of exponential decay by means of several numerical simulations, we can consider two damping β1ut and β2vt where β1,β2≥0. Then, making β2 go to zero, we can see a numerical sequence of eigenvalues converging to zero.We considerβ1=1 and we take β2 going to zero. Figures 1, 2, 3, and 4 show that when βi≠0, i=1,2, the real part of complex roots approaches negative asymptotes -βi/2, i=1,2.Figure 1
β
1
=
1, β2=1.Figure 2
β
1
=
1, β2=0.8.Figure 3
β
1
=
1, β2=0.6.Figure 4
β
1
=
1, β2=0.4.On the other hand, in Figures5 and 6, we can see a sequence of eigenvalues converging to 0 as β2→0.Figure 5
β
1
=
1, β2=0.04.Figure 6
β
1
=
1, β2=0.Next, we show results of numerical simulations of scheme (53)–(57) and its energy En. Here we use the stability criterion Δt≤Δx.


### 4.4. Undamped Case

Here, we considerα=0.25, L=1, T=2s, u(xj,0)=v(xj,0)=2sin(3πxj/L), ut(xj,0)=0, and vt(xj,0)=3sin(3πxj/L). At continuous level we obtain that E(0) ≈ 180,4029.Figures7, 8, 9, and 10 show the conservative behavior of numerical energy En to undamped system. We can see the conservative behavior of the energy En in agreement with Proposition 4.1. Moreover, it is clear that the accuracy of the approximations is achieved by means of refinements of type μ∶=Δt/Δx≤1.Figure 7
Δ
t
/
Δ
x
=
0.9600.Figure 8
Δ
t
/
Δ
x
=
0.8960.Figure 9
Δ
t
/
Δ
x
=
0.7680.Figure 10
Δ
t
/
Δ
x
=
0.7168.A simple analysis, by using absolute errorϵ:=|E(0)-E0| for each simulation, shows us the precision of the numerical energy. Table 1 shows the evolution of this error for each simulation.Table 1
μ
=
0.9600
μ
=
0.8960
μ
=
0.7680
μ
=
0.7168
ε
1.2103
0.2745
0.558
0.128


### 4.5. Full and Partial Damping

It is clear that considering two damping to the system (53)-(54) its numerical energy En decays exponentially. See Figures 11 and 13. Therefore, in this case the energy En is controllable by an exponential of type e-ωtn for some ω>0. On the other hand, taking into account damping only one dissipative mechanism we observe a slow decay from value ≈180 to value ≈90 (see Figures 12 and 14). We can note a lack of 50 by cent in terms of exponential decay.Figure 11
Δ
t
/
Δ
x
=
0.9600.Figure 12
Δ
t
/
Δ
x
=
0.960.Figure 13
Δ
t
/
Δ
x
=
0.8960.Figure 14
Δ
t
/
Δ
x
=
0.8960.Figure 15
Partial damping.Note that the numerical solutions of the functionv do not go to the state of rest. See Figures 16 and 18.Figure 16
Partial damping.Figure 17
Partial damping.Figure 18
Partial damping.


## 4.1. Explicit Time Integration Method

For our purpose we define a computational mesh definingΔx=L/(J+1) and Δt=T/(N+1) for J,N∈N and the nets given by
(52)x0=0<x1=Δx<⋯<xJ=JΔx<xJ+1=L,t0=0<t1=Δt<⋯<tN=NΔt<tN+1=T,
where xj=jΔx and tn=nΔt for j=0,1,2,…,J+1 and n=0,1,2,…,N+1.We consider the following finite-difference discretization of (48)-(49):
(53)ujn+1-2ujn+ujn-1Δt2-uj+1n-2ujn+uj-1nΔx2+ujn+1-ujn-12Δt+αvjn=0,(54)vjn+1-2vjn+vjn-1Δt2-vj+1n-2vjn+vj-1nΔx2+αujn=0,
for all j=1,2,…,J, n=1,2,…,N. Such scheme is in fact explicit and its implementation is straightforward. The boundary conditions are
(55)u0n=uJ+1n=0,v0n=vJ+1n=0,∀n=1,2,…,N,
and the initial conditions are given by
(56)uj0=u(xj,0),uj1=uj0+Δtut(xj,0),∀j=1,2,…,J,(57)vj0=v(xj,0),vj1=vj0+Δtvt(xj,0),∀j=1,2,…,J.The discretizations above are consistent with truncation error of second order inΔx and Δt. Therefore, it is immediate by Lax Lemma [11] that (53)-(54) converges. For convenience, we use the stability criterion (taking into account the conservative case) due to Courant-Friedrichs-Lewy; namely, Δt≤Δx.


## 4.2. Energy Conserving Scheme

In this section we will build the numerical energy to system (53)–(57). Following the techniques performed, for example, in [12, 13], this energy is given by
(58)En:=Δx2∑j=0J‍[(ujn+1-ujnΔt)2+(uj+1n+1-ujn+1Δx)(uj+1n-ujnΔx)+(vjn+1-vjnΔt)2+(vj+1n+1-vjn+1Δx)(vj+1n-vjnΔx)+α(vjnujn+1+ujnvjn+1)(ujn+1-ujnΔt)2].We note thatEn is the discrete version of the continuous energy (5) in one-dimensional setting. Moreover, one can show that En decreases. Instead of computing the time derivative of the energy we can use the summation by parts. The discrete energy En is an important numerical instrument to certify our analytical results concerning the polynomial stabilization of system (53)–(57).We have the following result.Theorem 6 (discrete energy).
Let(ujn,vjn) be a solution of the finite difference scheme (53)–(57). Then for all Δt and Δx, the discrete rate of change of energy of the numerical scheme (53)–(57) at the tn instant of time is given by
(59)En-En-1Δt=-∑j=1J‍(ujn+1-ujn-12Δt)2,
for all n=1,…,N,N+1.Proof.
We shortened the proof. It is identical to the proof of Proposition 1 in [12]. Then, we can write
(60)En+Δt∑j=1J‍(ujn+1-ujn-12Δt)2=En-1,
from where we have
(61)En≤E0,∀n=1,…,N,N+1.
Considering the conservative case we get(62)En=E0,∀n=1,2,…,N.


## 4.3. Discussion on Numerical Results

In this section, we consider two numerical approaches to certify the analytical results established in this work: the pseudospectral method to compute numerically the eigenvalues and also using the numerical energyEn in (58).For spectral analysis we apply the spectral method according to Trefethen’s book [14]. Using this numerical technique, we can see the numerical behavior of eigenvalues of (48)-(49). To see how this system is affected by a lack of exponential decay by means of several numerical simulations, we can consider two damping β1ut and β2vt where β1,β2≥0. Then, making β2 go to zero, we can see a numerical sequence of eigenvalues converging to zero.We considerβ1=1 and we take β2 going to zero. Figures 1, 2, 3, and 4 show that when βi≠0, i=1,2, the real part of complex roots approaches negative asymptotes -βi/2, i=1,2.Figure 1
β
1
=
1, β2=1.Figure 2
β
1
=
1, β2=0.8.Figure 3
β
1
=
1, β2=0.6.Figure 4
β
1
=
1, β2=0.4.On the other hand, in Figures5 and 6, we can see a sequence of eigenvalues converging to 0 as β2→0.Figure 5
β
1
=
1, β2=0.04.Figure 6
β
1
=
1, β2=0.Next, we show results of numerical simulations of scheme (53)–(57) and its energy En. Here we use the stability criterion Δt≤Δx.


## 4.4. Undamped Case

Here, we considerα=0.25, L=1, T=2s, u(xj,0)=v(xj,0)=2sin(3πxj/L), ut(xj,0)=0, and vt(xj,0)=3sin(3πxj/L). At continuous level we obtain that E(0) ≈ 180,4029.Figures7, 8, 9, and 10 show the conservative behavior of numerical energy En to undamped system. We can see the conservative behavior of the energy En in agreement with Proposition 4.1. Moreover, it is clear that the accuracy of the approximations is achieved by means of refinements of type μ∶=Δt/Δx≤1.Figure 7
Δ
t
/
Δ
x
=
0.9600.Figure 8
Δ
t
/
Δ
x
=
0.8960.Figure 9
Δ
t
/
Δ
x
=
0.7680.Figure 10
Δ
t
/
Δ
x
=
0.7168.A simple analysis, by using absolute errorϵ:=|E(0)-E0| for each simulation, shows us the precision of the numerical energy. Table 1 shows the evolution of this error for each simulation.Table 1
μ
=
0.9600
μ
=
0.8960
μ
=
0.7680
μ
=
0.7168
ε
1.2103
0.2745
0.558
0.128


## 4.5. Full and Partial Damping

It is clear that considering two damping to the system (53)-(54) its numerical energy En decays exponentially. See Figures 11 and 13. Therefore, in this case the energy En is controllable by an exponential of type e-ωtn for some ω>0. On the other hand, taking into account damping only one dissipative mechanism we observe a slow decay from value ≈180 to value ≈90 (see Figures 12 and 14). We can note a lack of 50 by cent in terms of exponential decay.Figure 11
Δ
t
/
Δ
x
=
0.9600.Figure 12
Δ
t
/
Δ
x
=
0.960.Figure 13
Δ
t
/
Δ
x
=
0.8960.Figure 14
Δ
t
/
Δ
x
=
0.8960.Figure 15
Partial damping.Note that the numerical solutions of the functionv do not go to the state of rest. See Figures 16 and 18.Figure 16
Partial damping.Figure 17
Partial damping.Figure 18
Partial damping.


## 5. Conclusions

In this work we considered a coupled system of two weakly dissipative wave equations. That is to say, the damping mechanism acts partially on the system and it is well known that this system is not exponentially stable. Our main contribution is twofold: the optimality of the polynomial decay in accordance with a recent theoretical contribution due to Borichev and Tomilov [6] and its numerical certification by using finite differences method. In particular, we derive a numerical energy and we show the lack of exponential decay by means of several numerical experiments.The analytical and numerical techniques performed in this work can be applied to systems where there exists a lack of exponential decay. See, for example, the work of Santos et al. for examples of systems with the property of lack of exponential decay and polynomial decay.



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*Source: 897080-2014-07-22.xml*