ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.14(2012) No.1,pp.38-47
Exponential Decay for a Transmission-Contact Problem with Frictional Damping Eugenio Cabanillas Lapa ∗ Instituto de Investigaci´on, Facultad de Ciencias Matem´aticas-UNMSM, Lima-Per´u (Received 27 November 2011, accepted 21 January 2012)
Abstract: In this article we study the evolution of displacement in a body constituted by two different types of materials: one part is simply elastic while the other has a frictional damping and may come into contact with a rigid foundation. Under this condition we have a transmission-contact problem. We prove that the solution of the problem decays exponentially to zero as time goes to infinity. Keywords: Contact problem; exponential decay;frictional damping M.S Classification A:35B40,35L70,45K05
1
Introduction
In this work we study the existence and the asymptotic behavior of weak solutions of the system ρ1 utt − buxx = 0
in ]0, L0 [×R+ ,
ρ2 vtt − avxx + αvt = 0 in ]L0 , L[×R+ , u(0, t) = 0 , u(L0 , t) = v(L0 , t), bux (L0 , t) = avx (L0 , t), t > 0,
(1)
v(L, t) ≤ g; vx (L, t) ≤ 0, (v(L, t) − g)vx (L, t) = 0, u(x, 0) = u0 (x), 0
v(x, 0) = v (x),
ut (x, 0) = u1 (x), 1
vt (x, 0) = v (x),
x ∈]0, L0 [, x ∈]L0 , L[,
where ρ1 , ρ2 are different densities of the material ; a, b are the elastic coefficients, α, g are positive constants and 0 < L0 < L . Equations (1) model the evolution of displacement in a elastic body consisting of two different types of materials one of them is simply elastic while, the other is subject to external forces and that may come into contact with the rigid foundation. The mathematical model which deals with the above situation is called transmission-contact problem. Here we are interested to study the resulting properties of a mixed material when one of its components is conservative and the other has a frictional damping and that come into contact with a rigid obstacle fixed at a distance g from the end x = L. The main question is about the asymptotic behavior;we may ask whether the sole dissipation produced by the frictional part is strong enough to produce an uniform rate of decay. The transmission problem have been a subject of intense studies from different mathematical points of view . The transmission problem to hyperbolic equations was studied by Dautray and Lions [5] who proved existence and regularity of solutions for the linear problem. While in Lions [10] was proved the exact controllability. Later, Lagnese [l1] extended this result; he showed the exact controllability for a class hyperbolic systems which include the transmission problem for homogeneous anisotropic materials. In [4] Cabanillas Lapa and Mu˜noz Rivera proved exponential decay for the system with time dependent coefficients. The exact controllability for the plate equations was proved by Liu and Williams [11]. For the memory condition on the boundary we can cite the works in Andrade , Fatori and Mu˜noz Rivera [1], Mu˜noz Rivera and Portillo Oquendo [15], Bae [3] and references therein. They used multiplier techniques and compactness arguments to ∗
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E. C. Lapa: Exponential Decay for a Transmission-Contact Problem with Frictional Damping
39
get exponential stability. Also, concerning to contact problems related to the evolution equations,we have very important results (see e.g. [2, 6, 13, 14, 16, 17] among others). They used perturbation techniques and the construction of suitable Lyapunov functionals to obtain exponential and polynomial decay. Some kind of Signorini transmission problem have been studied in [18]. Here, the authors used an iterative method to solve a variational inequality associated to the problem. The goal of this paper is to show that the solution of the transmission-contact problem decays exponentially to zero as time goes to infinity, no matter how small is the difference L − L0 . The main difficulties are that we have a more complicated situation involving transmission condition in x = L0 and Signorini’s contact condition in x = L , the dissipation only works in [L0 , L] and we need estimates over the whole domain [0, L]. The situation for the variational inequalities arising in transmission-contact problem is worse because we lead with weak solutions. So, we have neither regularity nor nice boundary conditions. We overcome this problem combining arguments of [4] and [13] and introducing suitable multipliers which allow us to control the energy only estimating over [L0 , L] . The rest of this article is developed as follows. In section 2 we present notations, preliminaries concepts, weak formulation and statement of the existence result for the penalized problem. In section 3 we obtain the solution of (1) as limit of solutions of the penalized problem. Finally, in section 4 we prove the exponential stability of the energy associated with the system.
Figure 1: Schematic of the elastic body consisting of two different types of materials and the rigid foundation
2
Penalized problem
Let us the following notations V = {(w, z) ∈ H 1 (0, L0 ) × H 1 (L0 , L) : w(L0 ) = z(L0 ), w(0) = 0} K = {(w, z) ∈ V : z(L) ≤ g}. To obtain the solution of penalized problem, we firstly define what we mean by a weak solution to (1). Definition 2.1 We say that the pair {u(x, t), v(x, t)} is a weak solution of (1) when {u, v} ∈ L∞ (0, T ; K) ∩ W 1,∞ (0, T ; L2 (0, L0 ) × L2 (L0 , L)) ∩ C 1,∞ (0, T ; H −1/2 (0, L0 ) × H −1/2 (L0 , L))
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International Journal of Nonlinear Science, Vol.14(2012), No.1, pp. 38-47
and satisfies ∫
L0
− ρ1 ∫
0
u1 (x)(φ(x, 0) − u0 )dx + ρ1 ⟨ut (T ), φ(T ) − u(T )⟩H −1/2 ×H 1/2
L
− ρ2 L0 ∫ T
v 1 (x)(ψ(x, 0) − v 0 )dx + ρ2 ⟨vt (T ), ψ(T ) − v(T )⟩H −1/2 ×H 1/2 ∫
0
∫
T
∫
L0 0
∫
T
∫
T
vt (ψt − vt ) dx dt+ 0
∫
T
∫
(2)
L0 L
ux (φx − ux ) dx dt + a ∫
L
ut (φt − ut ) dx dt − ρ2
0
b 0
∫
L0
− ρ1
vx (ψx − vx ) dx dt 0
L0
L
≥ −α
vt (ψ − v)dxdt 0
L0
for any {φ, ψ} ∈ L∞ (0, T ; K) ∩ W 1,∞ (0, T ; L2 (0, L0 ) × L2 (L0 , L)) To show the existence of weak solutions to problem (1), we follows similar ideas as in [6] . We first show existence of a strong solutions to the perturbed system ρ1 utt − buxx = 0
in ]0, L0 [×R+ ,
ρ2 vtt − avxx + αvt = 0
(3) +
in ]L0 , L[×R ,
u(0, t) = 0, t > 0, u(L0 , t) = v(L0 , t), bux (L0 , t) = avx (L0 , t), t > 0, 1 avx (L, t) = − [v(L, t) − g]+ − ϵvt (L, t) ϵ u(x, 0) = u0 (x), ut (x, 0) = u1 (x), x ∈]0, L0 [, 0
v(x, 0) = v (x),
1
vt (x, 0) = v (x),
x ∈]L0 , L[,
for any ϵ > 0. Letting ϵ → 0+ we will show that the corresponding limit is the weak solution to our original problem. Lemma 2.1 Let us denote by uk a sequence satisfying uk → u
weak star in L∞ (0, T ; H β (0, L))
ukt → ut
weak star in L∞ (0, T ; H α (0, L))
when k → +∞ and −1 ≤ α < β ≤ 1.Then for any r < β,we have uk → u strongly in C(0, T ; H r (0, L)) Proof. See [6]. To show the existence result, let us introduce the following functionals E(t) ≡ E(t, u, v) = E1 (t, u) + E2 (t, v). ∫ 1 L0 E1 (t, u) = (ρ1 |ut |2 + b|ux |2 )dx 2 0 ∫ 1 L E2 (t, v) = (ρ2 |vt |2 + a|vx |2 )dx 2 L0 The energy of the penalized problem (3)-(9) is Eϵ (t) = E(t, uϵ , v ϵ ) +
1 ϵ |[v (L, t) − g]+ |2 2ϵ
To simplify notation, we will avoid the superindex ϵ in our calculations.
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(4) (5) (6) (7) (8) (9)
E. C. Lapa: Exponential Decay for a Transmission-Contact Problem with Frictional Damping
41
Lemma 2.2 Let us take initial data satisfying {u0 , v 0 } ∈ (H 2 (0, L0 ) × H 2 (L0 , L)) ∩ V , {u1 , v 1 } ∈ V , and the compatibility condition bu0x (L0 ) = avx0 (L0 ), (10) 1 avx0 (L) = − [v 0 (L) − g]+ − ϵv 1 (L) ϵ Then there exists one solution {u, v} to (3)–(9) satisfying {u, v} ∈
2 ∩
W k,∞ (0, T ; H 2−k (0, L0 ) × H 2−k (L0 , L))
k=0
Proof. We employ the Faedo-Galerkin method. Indeed, we will choose a basis of V , B = {{φi , ψ i }, i ∈ N} such that {u0 , v 0 }, {u1 , v 1 } are in the span of {{φ0 , ψ 0 }, {φ1 , ψ 1 }}. 1 1 m Therefore, {um (0), v m (0)} = {u0 , v 0 } and {um t (0), vt (0)} = {u , v }. Standard results about ordinary differential equations guarantee that there exists only one solution
{u (t), v (t)} = m
m
m ∑
him (t){φi , ψ i }
i=1
of the approximated system ∫ L0 ∫ m i ρ1 utt φ dx + b 0
∫
∫
L0 i um x φx dx
0 L
+a L0
L m i vtt ψ dx
+ ρ2 L0
1 vxm ψxi dx + [v m (L, t) − g]+ ψ i (L) + ϵvtm (L, t)ψ i (L) + α ϵ
where i = 1, 2, . . . with initial data
∫
(11)
L
vtm ψ i dx = 0 L0
{um (0), v m (0)} = {u0 , v 0 },
(12)
m 1 1 {um t (0), vt (0)} = {u , v }.
On some interval [0, Tm [. To extend this solution to the whole interval [0, T ], for all T > 0 it is enough to show that approximated solutions are bounded independently of m and t. In order to do so , we first multiply equation (11) by h′jm (t) , integrating by parts and then summing up j, to obtain d m E (t) + α dt ϵ
∫
L
|vtm |2 dx = −ϵ|vtm (L, t)|2 .
(13)
L0
and integrating from 0 to t , we get ∫ t∫
∫
L
Eϵm (t) + α
t
|vtm |2 dxds + 0
L0
|vtm (L, s)|2 ds ≤ Eϵm (0).
(14)
0
Differentiating the approximated equation (11) with respect to t and then multiplying the resultanting equation by h′′jm (t), summing up in j, we obtain ∫ L d 1 m m 2 m m 2 E(t, um , v ) + α |vtt | dx = − [v m (L, t) − g]+ t vtt (L, t) − ϵ|vtt (L, t)| t t dt ϵ L0 (15) 1 m ϵ m + 2 2 ≤ − |vtt (L, t)| + 3 |[v (L, t) − g]t | . 2 ϵ m To get the second order estimate we have to show that E(0, um t , vt ) is bounded. Here we will use the special basis + chosen above. In fact, letting t → 0 in equation (11), multiplying the limit result by h′′jm (0), taking into account the compatibility conditions and using the transmission conditions we get ∫ L0 ∫ L ∫ L0 ∫ L ∫ L 2 m 2 0 m 0 m m ρ1 |um (0)| dx + ρ |v (0)| dx = b u u (0)dx + a v v (0)dx − α v 1 vtt (0)dx 2 tt tt xx tt xx tt 0
L0
0
L0
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L0
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International Journal of Nonlinear Science, Vol.14(2012), No.1, pp. 38-47
from which it follows that ∫
L0
∫ 2 |um tt (0)| dx
0
{∫
L m |vtt (0)|2 dx
+
≤c
L0
L0
∫ |u0xx |2 dx
0
}
L 0 2 (|vxx |
+
+ |v | )dx 1 2
(16)
L0
m From the above inequality we conclude that E(0, um t , vt ) is bounded. In this manner, combining (15) and (16) we derive that m E(t, um (17) t , vt ) is bounded .
Passage to limit as m → ∞. From the estimates (14) and (17) we can guarantee that there exists a subsequence of ({uϵm , v ϵm })m∈N still denoted in the same form, such that {uϵm , v ϵm } → {uϵ , v ϵ } weak ∗ in L∞ (0, T ; V ) ϵm ϵ ϵ ∞ 1 1 {uϵm t , vt } → {ut , vt } weak ∗ in L (0, T ; H (0, L0 ) × H (L0 , L)) ϵm ϵ ϵ ∞ 2 2 {uϵm tt , vtt } → {utt , vtt } weak ∗ in L (0, T ; L (0, L0 ) × L (L0 , L)).
Using Lemma 2.1 we have v ϵm (L, t) → v ϵ (L, t) uniformly on [0, T ] The remaining part of the proof is a matter of routine.
3
Contact problem
This section is devoted to obtain the solution for the system (1). We are interested in solving the problem as limit (when ϵ → 0) of the penalized problem. We first introduce the sets W = {(w, z) ∈ H 2 (0, L0 ) × H 2 (L0 , L) ∩ V : bwx (L0 ) = azx (L0 ), zx (L) = 0} . Wg = {(w, z) ∈ H 2 (0, L0 ) × H 2 (L0 , L) ∩ V : bwx (L0 ) = azx (L0 ), zx (L) = 0, z(L) = g} . VL = {(w, z) ∈ H 1 (0, L0 ) × H 1 (L0 , L) : w(L0 ) = z(L0 ), w(0) = 0 = z(L)} . We establish now the result that treats the existence of solutions for the transmission-contact problem associated with the wave equation. Theorem 3.1 Suppose that {u0 , v 0 } ∈ K, {u1 , v 1 } ∈ L2 (0, L0 ) × L2 (L0 , L) ,then there exists a weak solution of (1). Proof. Let us take a sequence {uϵ,0 , v ϵ,0 } ∈ W (or {uϵ,0 , v ϵ,0 } ∈ Wg when {u0 , v 0 } ∈ Wg ) and {uϵ,1 , v ϵ,1 } ∈ VL such that {uϵ,0 , v ϵ,0 } → {u0 , v 0 } in V {uϵ,1 , v ϵ,1 } → {u1 , v 1 } in L2 (0, L0 ) × L2 (L0 , L) { } when ϵ → 0.Observe that {uϵ,0 , v ϵ,0 }, {uϵ,1 , v ϵ,1 } satisfies the compatibility conditions (10). Let us denote by {uϵ , v ϵ } the strong solution of (3)-(9) with initial data {uϵ (x, 0), v ϵ (x, 0)} = {uϵ,0 (x), v ϵ,0 (x)} , {uϵt (x, 0), vtϵ (x, 0)} = {uϵ,1 (x), v ϵ,1 (x)}. Multiplying equation (3) by uϵt , equation (4) by vtϵ and performing an integration by parts we have d Eϵ (t) + α dt
∫
L
|vtϵ |2 = −ϵ|vtϵ (L, t)|2
(18)
L0
and integrating from 0 to t ,we get Eϵ (t) ≤ Eϵ (0).
(19)
Eϵ (t) is bounded
(20)
Thus, we conclude that
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E. C. Lapa: Exponential Decay for a Transmission-Contact Problem with Frictional Damping
43
which implies that {uϵ , v ϵ } → {u, v} weak ∗ in L∞ (0, T ; V ) {uϵt , vtϵ } → {ut , vt }weak ∗ in L∞ (0, T ; L2 (0, L0 ) × L2 (L0 , L)). Our next step is to show that {ut , vt } satisfies the variational inequality (2). To do this, let us multiply equation (3) by w − uϵ , equation (4) by z − v ϵ with {w, z} ∈ W 1,∞ (0, T ; L2 (0, L0 ) × L2 (L0 , L)) ∩ L∞ (0, T ; K) and performing an integration by parts we have ∫ L0 ∫ L0 − ρ1 uϵ,1 (x)(w(x, 0) − uϵ,0 )dx + ρ1 uϵt (T ), w(T ) − uϵ (T )dx ∫
0
− ρ2
v ∫
L0 T
ϵ,1
∫
0
T
∫
ϵ,0
L0
L0 T ∫ L
∫ uϵt (wt − uϵt ) dx dt − ρ2 ∫
T
vtϵ (zt − vtϵ ) dx dt+ ∫
0 L
vxϵ (zx − vxϵ ) dx dt 0
L0
∫
T
≥ −ϵ
∫
T
L
vt (L, t)(z(L, t) − v ϵ (L, t))dt − α
vtϵ (z − v ϵ )dxdt
0
0
L0
Now, following a similar procedure to that used in [16] we can show that ∫ T ∫ L0 ∫ T∫ lim sup {|uϵt |2 − |uϵx |2 }dxdt ≤ ϵ→0
and
0
∫
0
T
∫
0
ϵ→0
∫
L
T
L0
0
L0
{|ut |2 − |ux |2 }dxdt
0
∫
(22) L
{|vtϵ |2 − |vxϵ |2 }dxdt ≤
lim sup
(21)
L0
uϵx (wx − uϵx ) dx dt + a
0
∫
vtϵ (T ), z(T ) − v ϵ (T )dx
)dx + ρ2
0
b 0
(x)(z(x, 0) − v
L0
− ρ1 ∫
0 L
∫
L
{|vt |2 − |vx |2 }dxdt 0
L0
From (21)-(22) our conclusion follows.
4
Exponential decay
In this section we will prove that the solution of the transmission-contact problem (1) decays exponentially as time goes to infinity. To do it we first show that the energy associated to the penalized problem (3)–(9) decays exponentially as time goes to infinity. We used the perturbed energy method developed by Komornik and Zuazua [7] and the method of the integral inequalities introduced by Martinez [12], so we construct a suitable Lyapunov functional which allow us to get our result. From the convergence of solutions {uϵ , v ϵ } we obtain that the solution of the problem (1) decays exponentially too. For the sake of simplicity we omit the superscript ϵ. Note that (13) still holds d Eϵ (t) = −α|vt |2 − ϵ|vt (L, t)|2 . (23) dt and we have ∫ ∫ T
Eϵ (0) = Eϵ (T ) + α
T
|vt |2 dt + ϵ 0
|vt (L, t)|2 .
(24)
0
In the following lemmas we will prove some technical inequalities which will be useful to show the exponential decay of the solution. Lemma 4.1 The functional defined by
∫ J1 (t) =
L0
(x − L0 )ut ux dx
0
satisfies bLO d J1 (t) = −E1 (t, u) + dt 2
∫
LO
|ux (0, t)|2 .
0
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International Journal of Nonlinear Science, Vol.14(2012), No.1, pp. 38-47
Proof. Multiplying equation (3) by (x − L0 )ux and performing an integration in (0, L0 ) we get ∫ L0 ∫ L0 ρ1 (x − L0 )ux utt dx − b (x − L0 )ux uxx dx = 0. 0
Note that
(25)
0
d (x − L0 )ux ut = (x − L0 )ux utt + (x − L0 )uxt ut . dt
(26)
Now using (26) in (25) we get d ρ1 dt
∫
L0
∫
L0
(x − L0 )ux ut dx = ρ1
0
∫
0 L0
+b 0
1 d (x − L0 ) [ |ut |2 ] dx 2 dx
1 d (x − L0 ) [ |ux |2 ] dx 2 dx
Then, performing integration by parts and using the boundary conditions we get ∫ ∫ ∫ ρ 1 L0 b L0 d L0 ρ1 (x − L0 )ux ut dx. = − |ut |2 dx − |ux |2 dx dt 0 2 0 2 0 bL0 + |ux (0, t)|2 2 from which it follows that
d bL0 J1 (t) = −E1 (t, u) + |ux (0, t)|2 . dt 2
In order to control the punctual term |ux (0, t)|2 we implement the following lemma Lemma 4.2 Let us take q ∈ C 1 (0, L0 ) with q(0) > 0 and q(L0 ) = 0. Then, there exist positive constants C0 , N0 independent of initial data, such that the functional defined by ∫ L0 J2 (t) = N0 J1 (t) + qut ux dx 0
satisfies d J2 (t) ≤ −C0 E1 (t, u) dt Proof. Multiplying equation (3) by q ux and performing an integration in (0, L0 ) we get ∫ d L0 q ux ut dx dt 0 ∫ 1 L0 ′ bq(0) |ux (0, t)|2 , =− q ( ρ1 |ut |2 + b|ux |2 ) dx − 2 0 2 from which it follows that d dt
∫
L0
q ux ut dx ≤ −
0
bq(0) |ux (0, t)|2 + C1 E1 (t, u). 2
Adding N0 J1 (t) , to the above inequality, then taking N0 > C1 and choosing q(0) > N0 L0 we arrive at our result. Lemma 4.3 There exists a positive constant C2 , independent of initial data, such that the functional defined by ∫ L J3 (t) = (x − L0 )vt vx dx L0
satisfies d 1 (L − L0 ) J3 (t) ≤ − E2 (t, v) + I(L, t) + C2 dt 2 2
∫
L0
|vt |2 dx.
0
where I(L; t) = ρ2 |vt (L, t)|2 + a|vx (L, t)|2
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(27)
E. C. Lapa: Exponential Decay for a Transmission-Contact Problem with Frictional Damping
45
Proof. Multiplying equation (4) by (x − L0 )vx and performing an integration in (L0 , L) we get ∫ L ∫ L ∫ L0 ρ2 (x − L0 )vx vtt dx − (x − L0 )vx vxx dx + α (x − L0 )vx vt dx = 0. L0
L0
(28)
0
Performing integration by parts we get ∫ ∫ L0 d L (L − L0 ) (x − L0 )vx vt dx. = −E2 (t, v) + (ρ2 |vt (L)|2 + a|vx (L)|2 ) − α (x − L0 )vx vt dx dt L0 2 0 thus we conclude our proof. In order to control the punctual term I(L, t) we implement the following lemma Lemma 4.4 Let us take q ∈ C 1 (L0 , L) with q(L0 ) = 0 and q(L) < 0. Then, there exist positive constants C4 , C5 , α2 and N1 independent of initial data such that the functional defined by ∫ L J4 (t) = N1 J3 (t) + qvt vx dx L0
satisfies d 1 J4 (t) ≤ −C4 E2 (t, v) + (C5 + q(L)) I(L, T ) + α2 dt 2
∫
L
|vt |2 dx. L0
Proof. Multiplying equation (4) by q vx and performing an integration in (L0 , L), after simple calculations we get ∫ ∫ ∫ L d L 1 L ′ q(L) q vx vt dx = − q (ρ2 |vt |2 + a|vx |2 ) dx + (a|vx (L)|2 + ρ2 |vt (L)|2 ) − α qvx vt dx dt L0 2 L0 2 L0 from which, using Young’s inequality, it follows that ∫ ∫ L d L q(L) q vx vt dx ≤ (a|vx (L)|2 + ρ2 |vt (L)|2 ) + C3 E2 (t, v) + α1 |vt |2 dx. dt L0 2 L0 Then, we have
( ) d N1 N1 (L − L0 ) q(L) J4 (t) ≤ − ( − C3 ) E2 (t, v) + + I(L, t) dt 2 2 2 ∫ L + (α1 + N1 C2 ) |vt |2 dx. L0
Now taking N1 such that N1 > 2C3 , this lemma is proved. Theorem 4.5 Let us denote by {u, v} the solution of (3)–(9). Then there exist positive constants C and γ, such that Eϵ (t) ≤ C Eϵ (0)e−γ t . Proof. From (23), Lemmas (4.2) and (4.4) we get d {N Eϵ (t) + J2 (t) + J4 (t)} ≤ dt ∫ − C6 E(t, u, v) − (N α − α2 )
L
|vt |2 dx − N ϵ|vt (L, t)|2 + (N2 + q(L))I(L, t)
L0
for N > 0 . From which it follows C6 E(t, u, v) + a|vx (L, t)|2 ≤ ∫ L − (N α − α2 ) |vt |2 dx − N ϵ|vt (L, t)|2 + (N2 + 1 + q(L))I(L, t) L0
d − {N Eϵ (t) + J2 (t) + J4 (t)} dt
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International Journal of Nonlinear Science, Vol.14(2012), No.1, pp. 38-47
Now, using the inequality 1 a|vx (L, t)|2 = | − [v(L, t) − g]+ − ϵvt (L, t)|2 ϵ (0 α2 α−1 and observing that L(t) is equivalent to Eϵ (t) we arrive at ∫ T Eϵ (t)dt ≤ C (Lϵ (T ) + Lϵ (S)) C6 E(t, u, v) +
S
≤ CEϵ (S). for some C > 0. From the above inequality, our conclusion follows. We are in position to show the main result of this paper. Theorem 4.6 Let {u, v} be the solution of (1). Then there are exist positive constants C and γ independents of ϵ and t such that E(t) ≤ C E(0)e−γ t . Proof. We have from theorem 4.5, the convergence of {uϵ , v ϵ } and the lower semicontinuity of the energy that E(t, u, v) ≤ lim inf Eϵ (t) ≤ C{lim inf Eϵ (0)}e−γ t ≤ CE(0)e−γ t . ϵ→0
ϵ→0
This completes the proof.
Acknowledgements The authors are grateful to the anonymous referee for giving us valuable suggestions that improved our paper.
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E. C. Lapa: Exponential Decay for a Transmission-Contact Problem with Frictional Damping
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