QUARTERLY OF APPLIED MATHEMATICS
VOLUME LXVII, NUMBER 2
JUNE 2009, PAGES 249–263
S 0033-569X(09)01110-2
Article electronically published on March 19, 2009
ON THE STABILITY OF MINDLIN–TIMOSHENKO PLATES
By
´
HUGO D. FERNANDEZ
SARE
Department of Mathematics and Statistics, University of Konstanz, 78457, Konstanz, Germany
Abstract. We consider a Mindlin-Timoshenko model with frictional dissipations
acting on the equations for the rotation angles. We prove that this system is not exponentially stable independent of any relations between the constants of the system, which
is different from the analogous one-dimensional case. Moreover, we show that the solution decays polynomially to zero, with rates that can be improved depending on the
regularity of the initial data.
1. Introduction. The conservative Mindlin-Timoshenko model in the two-dimensional case is given by
∂
∂w
∂
∂w
ψ+
+
ϕ+
= 0,
(1.1)
ρhwtt − K
∂x
∂x
∂y
∂y
2
∂ ψ 1 − µ ∂2ψ 1 + µ ∂2ϕ
∂w
ρh3
ψtt − D
+
+
+K ψ+
= 0, (1.2)
12
∂x2
2 ∂y 2
2 ∂x∂y
∂x
2
∂ ϕ 1 − µ ∂2ϕ 1 + µ ∂2ψ
ρh3
∂w
ϕtt − D
+
+
+
K
ϕ
+
= 0, (1.3)
12
∂y 2
2 ∂x2
2 ∂x∂y
∂y
where Ω ⊂ R2 is bounded, ρ is the (constant) mass per unit of surface area, h is the
(uniform) plate thickness, µ is Poisson’s ratio (0 < µ < 12 in physical situations), D is
the modulus of flexural rigidity, and K is the shear modulus. The functions w, ψ and
ϕ depend on (t, x, y) ∈ [0, ∞) × Ω, where w models the transverse displacement of the
plate, and ψ, ϕ are the rotation angles of a filament of the plate; cp. [7, 8].
The main difference of this system to the analogous one-dimensional case (ϕ ≡ 0) is
that here another equation for rotation angles is considered. Note also that the coupling
between the equations of the rotational angles (ψ, ϕ) and the displacement equation w
is weaker than in one dimension. That is, while in two dimensions the coupling is given
by the gradient of the functions (see system (1.8)–(1.9)), in the one-dimensional case
Received September 17, 2007.
2000 Mathematics Subject Classification. Primary 35B40, 74H40.
Key words and phrases. Timoshenko plates, non-exponential stability, polynomial stability.
The author was supported by the DFG-project “Hyperbolic Thermoelasticity” (RA 504/3-3).
E-mail address: [email protected]
c
2009
Brown University
Reverts to public domain 28 years from publication
249
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250
HUGO D. FERNÁNDEZ SARE
the coupling is given by partial derivatives ψx and ϕy in (1.1), wx in (1.2), and wy
in (1.3). Therefore the questions of how it is possible to stabilize the system and find
“sufficient” dissipations to produce exponential stability are interesting, but they are not
much studied in the literature.
In [7], Lagnese considered a bounded domain Ω having a Lipschitz boundary Γ such
that Γ = Γ0 ∪ Γ1 , where Γ0 and Γ1 are relatively open, disjoints subsets of Γ with Γ1 = ∅.
He considered the following boundary conditions:
w=ψ=ϕ
∂w
∂w
K
+ ψ,
+ϕ ·ν
∂x
∂y
1 − µ ∂ϕ ∂ψ
,
+
·ν
2
∂x
∂y
∂ψ
∂ψ
∂ϕ
+
+µ
,
·ν
∂y
∂y
∂x
∂ψ
∂ϕ
+µ
∂x
∂y
1 − µ ∂ϕ
D
2
∂x
D
=
0
in Γ0 ,
(1.4)
=
m1
in Γ1 ,
(1.5)
=
m2
in Γ1 ,
(1.6)
=
m3
in Γ1 ,
(1.7)
where ν := (ν1 , ν2 ) is the unit exterior normal to Γ = ∂Ω and {m1 , m2 , m3 } are the linear
boundary feedbacks given by
{m1 , m2 , m3 } = −F {wt , ψt , ϕt },
with F = [fij ] a 3 × 3 matrix of real L∞ (Γ1 ) functions such that F is symmetric and
positive semidefinite on Γ1 . In that condition, Lagnese proved that the system (1.1)–(1.3)
is exponentially stable, without any restrictions on the coefficients of the system. The
same result is obtained by Muñoz Rivera and Portillo Oquendo [10], where, in (1.5)–(1.7),
they consider boundary dissipations of memory type; that is,
t
∂w
∂w
(s) + ψ(s),
(s) + ϕ(s) · νds = 0 in Γ1 ,
w + K g1 (t − s)
∂x
∂y
0
t
∂ψ(s)
∂ϕ(s) 1 − µ ∂ϕ(s) ∂ψ(s)
+µ
,
+
ψ + D g2 (t − s)
· νds = 0 in Γ1 ,
∂x
∂y
2
∂x
∂y
0
t
1 − µ ∂ϕ(s) ∂ψ(s)
∂ϕ(s)
∂ψ(s)
ϕ + D g3 (t − s)
+
,
+µ
· νds = 0 in Γ1 .
2
∂x
∂y
∂y
∂x
0
With these boundary feedbacks, together with condition (1.4), they proved that the
solutions of the system (1.1)–(1.3) are exponentially stable provided that the kernels
have exponential behavior and are polynomially stable for kernels of polynomial type.
Similar dissipations are used by De Lima Santos [3], where the author considered a
Timoshenko model in Ω ⊂ Rn .
In this work, we are interested in introducing another type of dissipation. For example,
taking into account the papers mentioned above, if we consider three frictional internal
dissipations into the system; that is, introducing the terms wt in (1.1), ψt in (1.2), and ϕt
in (1.3), the exponential behavior of the solutions of the system is easily obtained. The
natural questions are the following: what happens if we remove one of these dissipations
and, of course, which is the “natural” candidate to be removed? Looking at the onedimensional model we can deduce some conclusions in order to solve these questions. In
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ON THE STABILITY OF MINDLIN–TIMOSHENKO PLATES
251
[12, 14], the authors considered the one-dimensional damped Timoshenko system
ρ1 ϕtt − k(ϕx + ψ)x = 0
in (0, L),
(1.8)
ρ2 ψtt − bψxx + k(ϕx + ψ) + dψt = 0
in (0, L),
(1.9)
and proved that the solution of the system is exponentially stable if and only if the wave
speeds of the system are equal, that is, if
ρ1
ρ2
= .
(1.10)
k
b
The Timoshenko model (1.8)–(1.9) with several types of dissipations has been studied by
many authors; see for example [4, 5, 6, 11, 12, 14] and the references therein. The common
point in almost every work is the condition (1.10), necessary and sufficient to obtain
exponential stability. Note that, removing the dissipative term dψt in (1.9) and putting
ϕt in (1.8), we can deduce that the system is not exponentially stable independently if
(1.10) holds or not. From these observations for the system (1.8)–(1.9) we can establish
an equivalent problem in two dimensions. That is, introducing frictional dissipations in
the rotational angle equations (1.2)–(1.3), we obtain a new dissipative system where the
rate of decay for the solutions of that system appears as an open problem to be analyzed.
In other words, the purpose of this paper is to study the stability of the Timoshenko
system
∂w
∂w
,ϕ +
= 0,
(1.11)
ρ1 wtt − Kdiv ψ +
∂x
∂y
∂ψ
∂ϕ 1 − µ ∂ϕ ∂ψ
∂w
+µ ,
+
(1.12)
ρ2 ψtt − Ddiv
+K ψ+
+ d1 ψt = 0,
∂x
∂y
2
∂x
∂y
∂x
1 − µ ∂ϕ ∂ψ
∂ψ
∂ϕ
∂w
ρ2 ϕtt − Ddiv
+
+µ
(1.13)
,
+K ϕ+
+ d2 ϕt = 0,
2
∂x
∂y
∂y
∂x
∂y
3
where ρ1 := ρh, and ρ2 := ρh
12 in the system (1.1)–(1.3). We will prove that the
system (1.11)–(1.13) is not exponentially stable, independent of any relation between
the constants of the system. This leads to a different result from the one obtained in
[12] in the one-dimensional case, where the condition (1.10) was sufficient and necessary
to obtain exponential stability. Moreover, using multiplier techniques and Prüss et al.’s
result [13], we will prove that the system (1.11)–(1.13) is polynomially stable with rates
that can be improved depending on the initial data. We would like to add here that
this is the first time that the asymptotic behavior of the system (1.11)–(1.13) has been
studied, and that our analysis clearly shows the differences in the dimensions.
The paper is organized as follows: In Section 2 we shall look at the existence and
uniqueness results using semigroup theory. The Timoshenko system (1.11)–(1.13) is
shown to be not exponentially stable subject to mixed boundary conditions in Section 3.
Finally, in Section 4 we study the polynomial stability of the system (1.11)–(1.13) with
Dirichlet boundary conditions.
2. Existence and uniqueness. We will use the standard notation H k (Ω) or H0k (Ω)
to denote usual Sobolev spaces of order k over the regular domain Ω, and set L2 (Ω) =
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252
HUGO D. FERNÁNDEZ SARE
H 0 (Ω). We consider the Timoshenko system (1.11)–(1.13) with the following Dirichlet
boundary conditions:
w(x, t) = ψ(x, t) = ϕ(x, t) = 0 in
∂Ω × R+ ,
(2.1)
and initial conditions
w(x, 0) = w0 (x) ,
wt (x, 0) = w1 (x) ,
in Ω,
ψ(x, 0) = ψ0 (x) ,
ψt (x, 0) = ψ1 (x) ,
in Ω,
ϕ(x, 0) = ϕ0 (x) ,
ϕt (x, 0) = ϕ1 (x) ,
in Ω.
(2.2)
In order to obtain existence, uniqueness, and stability results, we will use semigroup
theory. For this purpose we rewrite the system an as evolution equation for
U = (w, wt , ψ, ψt , ϕ, ϕt ) ≡ (u1 , u2 , u3 , u4 , u5 , u6 , ) .
Then U formally satisfies
Ut = A1 U,
U (0) = U0 ,
where U0 = (w0 , w1 , ψ0 , ψ1 , ϕ0 , ϕ1 ) , and A1 is the (yet formal) differential operator
⎛
⎞
0
Id
0
0
0
0
⎜
⎟
⎜ K
⎟
K
K
0
∂
0
∂
0
⎜ ρ1 ∆
⎟
x
y
ρ
ρ
1
1
⎜
⎟
⎜
⎟
⎜
⎟
0
0
0
Id
0
0
⎜
⎟
(2.3)
A1 := ⎜
⎟,
d1
D 1+µ
2
⎜ − K ∂x 0
⎟
∂
B
−
Id
0
1
xy
⎜ ρ2
⎟
ρ2
ρ2
2
⎜
⎟
⎜
⎟
0
0
0
0
0
Id ⎟
⎜
⎝
⎠
2
d2
∂
0
B
−
Id
− ρK2 ∂y 0 ρD2 1+µ
2
xy
2
ρ2
where the differential operators Bi (i = 1, 2), are defined by
1−µ
D 2
k
B1 =
∂ +
∂y2 − Id,
ρ2 x
2
ρ2
1−µ
D
k
2
2
B2 =
∂x + ∂y − Id.
ρ2
2
ρ2
Let
H1 := H01 (Ω) × L2 (Ω) × H01 (Ω) × L2 (Ω) × H01 (Ω) × L2 (Ω)
be the Hilbert space. In order to endow the space H1 with a norm associated to the
energy of the system (1.11)–(1.13), we will use the following result.
Lemma 2.1. There exists α0 > 0 such that, for all (ψ, ϕ) ∈ [H01 (Ω)]2 ,
1−µ
2
2
|ψx | + |ϕy | +
|ψy + ϕx |2 + µψx ϕy + µϕy ψ x dxdy
2
Ω
≥ α ||ψ||2H 1 + ||ϕ||2H 1 .
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ON THE STABILITY OF MINDLIN–TIMOSHENKO PLATES
253
Moreover, for every K0 > 0 there exists β := β(K0 ) > 0 such that for all K ≥ K0 and
for all (w, ψ, ϕ) ∈ [H01 (Ω)]3 ,
1−µ
|ψx |2 + |ϕy |2 +
|ψy + ϕx |2 + µψx ϕy + µϕy ψ x dxdy
2
Ω
+ K |ψ+wx |2 dxdy+K |ϕ + wx |2 dxdy ≥ β ||∇ψ||2L2 +||∇ϕ||2L2 +||∇w||2L2 .
Ω
Ω
Proof. It is a direct consequence of Korn’s Inequality; see [7].
Then, using the previous Lemma, we can obtain that
||U ||2H1
= ||(u1 , u2 , u3 , u4 , u5 , u6 )||2H1
= ρ1 ||u2 ||2L2 + ρ2 ||u4 ||2L2 + ρ2 ||u6 ||2L2 + D||u3x ||2L2 + D||u5y ||2L2
+K||u3 + u1x ||2L2 + K||u5 + u1y ||2L2 + µ(u3x , u5y )L2 + µ(u5y , u3x )L2 (2.4)
is equivalent with the usual norm in H1 .
Therefore, it is not difficult to prove that the operator A1 is maximal-dissipative; that
is, A1 is the infinitesimal generator of a C0 contraction semigroup on H1 . Thus, we have
the following result about existence and uniqueness of solutions.
Theorem 2.2. Let U0 = (w0 , w1 , ψ0 , ψ1 , ϕ0 , ϕ1 ) ∈ H1 . Then there exists a unique
solution U (t) = (w, wt , ψ, ψt , ϕ, ϕt ) to the system (1.11)–(1.13) with Dirichlet boundary
conditions (2.1) satisfying
U ∈ C(R+ ; D(A1 )) ∩ C 1 (R+ ; H1 ).
Moreover, if U0 ∈ D(An1 ), then
U ∈ C n−k (R+ ; D(Ak1 )),
k = 0, 1, · · ·, n.
Remark 2.3. The same analysis can be applied to obtain existence and uniqueness
results for mixed boundary conditions.
3. Non-exponential stability. In this section we will prove that the system (1.11)–
(1.13) is not exponentially stable for suitable boundary conditions. In fact, we consider
Ω ⊂ R2 as the rectangle
Ω := [0, L1 ] × [0, L2 ] ,
with L1 , L2 > 0.
We define the sets
Γ1
Γ2
(x, y) : 0 < x < L1 , y = 0, L2 ,
:=
(x, y) : 0 < y < L2 , x = 0, L1 .
:=
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254
HUGO D. FERNÁNDEZ SARE
Note that Γ := ∂Ω = Γ1 ∪ Γ2 . The boundary conditions considered for the system
(1.11)–(1.13) are the following:
w=0
in Γ,
∂ψ
1 − µ ∂ϕ ∂ψ
∂ϕ
+
+µ
,
· ν = 0 in Γ1 ,
2
∂x
∂y
∂y
∂x
∂ψ
∂ϕ 1 − µ ∂ϕ ∂ψ
+µ
,
+
· ν = 0 in Γ2 ,
∂x
∂y
2
∂x
∂y
ψ = 0,
ϕ = 0,
(3.1)
where ν := (ν1 , ν2 ) is the unit exterior normal to Γ = ∂Ω. Therefore, the semigroup
formulation is given in the Hilbert space,
H2 := H01 (Ω) × L2 (Ω) × HΓ11 (Ω) × L2 (Ω) × HΓ12 (Ω) × L2 (Ω),
where
HΓ1i (Ω) :=
u ∈ H 1 (Ω) : u = 0 in Γi
(i = 1, 2)
and with the same norm given by (2.4).
We shall use the following well-known result from semigroup theory (see, e.g., [9,
Theorem 1.3.2]).
Lemma 3.1. A semigroup of contractions {etA }t≥0 in a Hilbert space with norm · is
exponentially stable if and only if
(i) the resolvent set (A) of A contains the imaginary axis
and
(ii) lim sup (iλId − A)−1 < ∞
λ→±∞
hold.
Hence it suffices to show the existence of sequences (λn )n ⊂ R with
lim |λn | = ∞,
n→∞
and (Un )n ⊂ D(A1 ), (Fn )n ⊂ H, such that
(iλn Id − A1 )Un = Fn
lim Un H1 = ∞.
is bounded and
n→∞
As Fn ≡ F we choose F := (0, f 2 , 0, f 4 , 0, f 6 ) with
f2
:= F 2 sin(δλ1 x) sin(δλ2 y),
F 2 = 0 (constant),
f4
:= F 4 cos(δλ1 x) sin(δλ2 y),
F 4 = 0 (constant),
f6
:= F 6 sin(δλ1 x) cos(δλ2 y),
F 6 = 0 (constant),
where
λ1 ≡ λ1,n
nπ
:=
,
δL1
λ2 ≡ λ2,n
Finally we define
λ ≡ λn :=
1
2
3
4
5
nπ
:=
δL2
(n ∈ N) ,
δ :=
λ21 + λ22 .
6 The solution U = (v , v , v , v , v , v ) of the resolvent equation
(iλId − A1 )U = F
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ρ1
.
k
(3.2)
ON THE STABILITY OF MINDLIN–TIMOSHENKO PLATES
255
should satisfy
iλv 1 − v 2
iλv 2 −
iλv 4 −
iλv 6 −
D 3
v +
ρ2 xx
D
ρ2
1−µ
2
1−µ
2
3
+
vyy
5
5
+ vyy
+
vxx
k 3
k
(v + vx1 )x − − (v 5 + vy1 )y
ρ1
ρ1
1+µ
2
1+µ
2
5
vxy
3
vxy
= 0,
= f 2,
iλv 3 − v 4
d1
k
+ (v 3 + vx1 ) + v 4
ρ2
ρ2
= 0,
iλv 5 − v 6
d2
k
+ (v 5 + vy1 ) + v 6
ρ2
ρ2
= 0,
= f 4,
= f 6.
(3.3)
Eliminating v 2 , v 4 , v 6 we obtain the following system for v 1 , v 3 , v 5 :
−λ2 ρ1 v 1 − k(v 3 + vx1 )x − k(v 5 + vy1 )y
3
5 3
−λ2 ρ2 v 3 − D vxx
vyy + 1+µ
vxy + k(v 3 + vx1 ) + iλd1 v 3
+ 1−µ
2
2
5
3 5
−λ2 ρ2 v 5 − D 1−µ
vxx + vyy
vxy + k(v 5 + vy1 ) + iλd2 v 5
+ 1+µ
2
2
= ρ1 f 2 ,
= ρ2 f 4 ,
(3.4)
= ρ2 f 6 .
System (3.4) can be solved by
v 1 (x, y) := A sin(δλ1 x) sin(δλ2 y),
v 3 (x, y) := B cos(δλ1 x) sin(δλ2 y),
v 5 (x, y) := C sin(δλ1 x) cos(δλ2 y),
where A, B, C depend on λ and will be determined explicitly in the sequel. Note that this
choice is just compatible with the boundary conditions (3.1). System (3.4) is equivalent
to finding A, B, C such that
− λ2 ρ1 A + kδ 2 λ21 + λ22 A + kδλ1 B + kδλ2 C = ρ1 F 2 , (3.5)
1−µ
1+µ
2
2 2
2 2
−λ ρ2 B + Dδ λ1 B + D
δ λ2 B + D
δ 2 λ1 λ2 C
2
2
+kB + kδλ1 A + iλd1 B
1−µ
1+µ 2
−λ2 ρ2 C + D
δ 2 λ21 C + Dδ 2 λ22 C + D
δ λ1 λ2 B
2
2
= ρ2 F 4 , (3.6)
+kC + kδλ2 A + iλd2 C
= ρ2 F 6 . (3.7)
Using the definitions of δ and λ, we obtain from (3.5) that
1
λ2
C + δF 2
λ1
λ1
(3.8)
λ1
1
B + δF 2 .
λ2
λ2
(3.9)
B=−
or
C=−
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256
HUGO D. FERNÁNDEZ SARE
Using (3.9) in (3.6) we obtain
1+µ
1−µ
−λ2 ρ2 − Dδ 2
+ k + iλd1 B + kδλ1 A = ρ2 F 4 − D
δ 3 λ1 F 2 .
2
2
(3.10)
Similarly, using (3.8) in (3.7) we obtain
1+µ
1−µ
2
2
6
−λ ρ2 − Dδ
+ k + iλd2 C + kδλ2 A = ρ2 F − D
δ 3 λ2 F 2 .
2
2
(3.11)
Let
ρ1 1 − µ
Θ := ρ2 − D
.
(3.12)
k
2
Then, using the definition of δ, we obtain from (3.10)–(3.11) that A, B satisfies
2
1+µ
(3.13)
−λ Θ + k + iλd1 B + kδλ1 A = ρ2 F 4 − D
δ 3 λ1 F 2 ,
2
2
1+µ
−λ Θ + k + iλd2 C + kδλ2 A = ρ2 F 6 − D
(3.14)
δ 3 λ2 F 2 .
2
Remark 3.2. Note that the condition Θ = 0 in (3.12) gives a relationship (in the
2-dimensional case) similar to the relation
ρ2
ρ1
= ,
k
b
which is a necessary and sufficient condition for exponential stability in the 1-dimensional
case; see [12]. We will show that the system (1.11)–(1.13) is non-exponentially stable
and independent of any relation between the coefficients of the system, in particular of
Θ = 0 in (3.12).
Using (3.9) in (3.14) we obtain
λ21
2
1+µ
λ1
6
δ 3 λ1 F 2
−λ Θ + k + iλd2 2 B − kδλ1 A = − ρ2 F + D
λ2
λ2
2
+ −λ2 Θ + k + iλd2 δλ1 F 2 .
(3.15)
Then, adding the equalities (3.13) and (3.15) yields
2
2
λ21
λ1
−λ Θ + k + iλd1 + −λ Θ + k + iλd2 2 B = ρ2 F 4 − ρ2 F 6
λ2
λ2
2
+ −λ Θ + k + iλd2 δλ1 F 2 .
This is
B
=
λ1
ρ2 F 4 − ρ2 F 6 + −λ2 Θ + k + iλd2 δλ1 F 2
λ2
λ21
λ21
λ21
2
−λ Θ 1 + 2 + k 1 + 2 + iλ d1 + d2 2
λ2
λ2
λ2
and using (3.13) we have
A
=
ρ2
F4 − D
kδλ1
1+µ
2
δ2 2 2
F − −λ Θ + k + iλd1 B
k
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(3.16)
(3.17)
ON THE STABILITY OF MINDLIN–TIMOSHENKO PLATES
257
with B given by (3.16). We define
λ2
λ2
λ2
Q(λ) := −λ2 Θ 1 + 12 + k 1 + 12 + iλ d1 + d2 12 ,
λ2
λ2
λ2
where, using the definitions of λi (i = 1, 2), we can conclude that
L :=
λ1
L2
=
> 0.
λ2
L1
(3.18)
Therefore Q(λ) is given by
Q(λ) = −λ2 Θ 1 + L2 + k 1 + L2 + iλ d1 + d2 L2 .
(3.19)
We also define the following functions:
6
1+µ
ρ2 4
1
4
2
2
F +
A1 (λ) :=
,
(3.20)
kρ2 LF − F − D
δ 1+L
kδλ1
Q(λ)
2
1
− λ4 λ1 Θ2 δF 2 + iλ3 λ1 Θ (d1 − d2 ) δF 2 + λ2 λ1 (d2 d1 + 2Θk) δF 2
A2 (λ) :=
Q(λ)
1 + µ δ2
4
6
(1 + L2 ) − iλλ1 (d1 + d2 ) kδF 2
+λ Θ (F − LF )ρ2 + D
2
k
1+µ
6
4
2
2
+iλ d1 ρ2 (LF − F ) − D
(3.21)
(d1 + d2 L ) − δλ1 kF .
2
2
Then we have in (3.17) that
A = A1 (λ) + A2 (λ).
Recalling that
v 2 = iλv 1 = iλA sin(δλ1 x) sin(δλ2 y)
we get
v 2 = iλA1 (λ) + iλA2 (λ) sin(δλ1 x) sin(δλ2 y).
Note that
||Un ||H
≥ ||v 2 ||L2
1/2
=
|v 2 |2 dxdy
Ω
≥ −C1 |λA1 (λ)| + C2 |λA2 (λ)| ,
where Ci := Ci (L1 , L2 ) > 0, i = 1, 2. Then, to complete our result, it is sufficient to
show that
(i) The sequence {λA1 (λ)}λ ⊂ R+ is bounded, and
(ii) |λA2 (λ)| → ∞ as λ → ∞, independent of any relation between the constants of
the system.
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258
HUGO D. FERNÁNDEZ SARE
In fact, using the definitions of λ, λ1 in (3.20) we obtain
λA1 (λ) =
ρ2
kδ
1+µ
δ 2 1 + L2
1
2
1 + 2F4 +
.
k
L
1 + L2 + i d1 + d2 L2
−λΘ 1 + L2 +
λ
kρ2 LF 6 − F 4 − D
Then {λA1 (λ)}λ is bounded, which completes the proof of item (i). On the other hand,
note that item (ii) is obvious in the case Θ = 0. When Θ = 0 we have in (3.21) that
1
2λ3 λ1 d2 d1 δF 2 − iλ2 λ1 (d1 + d2 ) kδF 2
λA2 (λ) =
Q0 (λ)
1+µ
6
4
2
2
+iλ d1 ρ2 (LF − F ) − D
(d1 + d2 L ) − δλλ1 kF ,
2
2
with
Q0 (λ) = k(1 + L2 ) + iλ(d1 + d2 L2 ).
Therefore |λA2 (λ)| −→ ∞. Thus we have proved the following theorem.
Theorem 3.3. The Timoshenko system (1.11)–(1.13) with boundary conditions (3.1)
is not exponentially stable, independent of any relation between the constants of the
system.
Remark 3.4. As in the 1-dimensional case, the non-exponential stability in Dirichlet
boundary conditions (2.1) is still an open problem. Note also that the function that
generates the non-exponential stability, that is {λA2 (λ)}λ , has the behavior as |λA2 (λ)| ∼
◦(λ3 ), which produces the expectation that, to show polynomial stability results, we will
need energies of higher order; see Section 4.
4. Polynomial stability. In this section we shall prove that the system (1.11)–
(1.13) with boundary conditions (2.1) is polynomially stable. The energy of first order
associated to the system (1.11)–(1.13) is given by
E1 (t) := E1 (t; w, ψ, ϕ)
1 ρ1 |wt |2 +ρ2 |ψt |2 +ρ2 |ϕt |2 +K|ψ + wx |2 +K|ϕ+wx |2 + D|ψx |2
=
2 Ω
1−µ
2
+ D|ϕy | +
(4.1)
D|ψy +ϕx |2 +2Dµψx ϕy dxdy,
2
which is obtained multiplying equation (1.11) by wt , (1.12) by ψt , and (1.13) by ϕt . Also,
we can define the energies
(i)
(i)
(i)
Ei+1 (t) := E1 (t; ∂t w, ∂t ψ, ∂t ϕ) ,
i = 1, 2, 3.
It is not difficult to show that
d
(i)
(i)
Ei (t) = −d1 |∂t ψ|2dxdy − d2 |∂t ϕ|2dxdy ,
dt
Ω
Ω
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i = 1, 2, 3, 4.
(4.2)
(4.3)
ON THE STABILITY OF MINDLIN–TIMOSHENKO PLATES
We define
F1 (t) :=
[ρ1 wt w + ρ2 ψt ψ + ρ2 ϕt ϕ +
Ω
259
d1 2 d2 2
ψ + ϕ ] dxdy.
2
2
(4.4)
Then, multiplying equation (1.11) by w, (1.12) by ψ, and (1.13) by ϕ, results in
d
1−µ
2
2
|ψx | + |ϕy | +
F1 (t) = −D
|ψy + ϕx |2 + 2µψy ϕx dxdy
dt
2
Ω
−K |ψ + wx |2 dxdy − K |ϕ + wx |2 dxdy + ρ2 |ψt |2 dxdy
Ω
+ρ2
Ω
|ϕt |2 dxdy + ρ1
Ω
Ω
|wt |2 dxdy.
(4.5)
Ω
Let q : Ω → R defined by q(x, y) = x. We define
1−µ
F2 (t) := −D
ψxt + µϕyt ,
(ψyt − ϕxt ) .∇wq(x, y) dxdy.
2
Ω
(4.6)
Then, differentiating equation (1.12) with respect to t and multiplying by q(x, y)wt in
L2 (Ω) results in
d
F2 (t) = −K |wt |2dxdy + ρ2 ψttt q(x, y)wt dxdy + K ψt q(x, y)wt dxdy
dt
Ω
Ω
Ω
+d1 ψtt q(x, y)wt dxdy + D (ψxt + µϕyt ) wt dxdy
Ω
Ω
1−µ
−D
ψxtt + µϕytt ,
(ψytt − ϕxtt ) .∇wq(x, y) dxdy,
2
Ω
where we can conclude that there exists
Ci := Ci (ρ1 , ρ2 , K, D, µ, Ω) > 0 ,
i = 1, 2,
(4.7)
such that
d
K
F2 (t) ≤ −
|ψtt |2 + |ψttt |2 dxdy
|wt |2dxdy + C1 ||ψt ||2H 1 + ||ϕt ||2H 1 + C2
dt
2 Ω
Ω
1−µ
−D
(4.8)
ψxtt + µϕytt ,
(ψytt − ϕxtt ) .∇wq(x, y) dxdy.
2
Ω
We will use the letter C to denote several positive constants defined as in (4.7). Defining
F3 (t) := F1 (t) +
4ρ1
F2 (t),
K
(4.9)
and using (4.5) and (4.8) we have
d
F3 (t) ≤ −2E1 (t) + C ||ψt ||2H 1 + ||ϕt ||2H 1 + C
|ψtt |2 + |ψttt |2 dxdy
dt
Ω
1−µ
4ρ1 D
−
ψxtt + µϕytt ,
(ψytt − ϕxtt ) .∇wq(x, y) dxdy.
K Ω
2
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(4.10)
260
HUGO D. FERNÁNDEZ SARE
Remark 4.1. In (4.10) we already have the first order energy with negative signs,
but it is necessary to estimate other higher-order terms. The following functions will be
defined in order to estimate these terms.
First, note that using Korn’s Inequality [2], we have that there exist constants α, β > 0
such that (see [7])
1−µ
|ψx |2 + |ϕy |2 +
|ψy + ϕx |2 + 2µψx ϕy dxdy ≥ α ||ψ||2H 1 + ||ϕ||2H 1 (4.11)
2
Ω
and
1−µ
2
2
|ψx | + |ϕy | +
|ψy + ϕx |2 + 2µψx ϕy dxdy
2
Ω
+K |ψ + wx |2 dxdy + K |ϕ + wx |2 dxdy ≥ β ||∇ψ||2L2 + ||∇ϕ||2L2 + ||∇w||2L2 .
Ω
Ω
(4.12)
On the other hand, differentiating equations (1.12)–(1.13) with respect to t and multiplying by ψt and ϕt respectively, results in
d
ρ2 [ψtt ψt + ϕtt ϕt ] dxdy
dt Ω
1−µ
= −D |ψxt |2 +|ϕyt |2 +
|ψyt +ϕxt |2 +2µψyt ϕxt dxdy
2
Ω
+ ρ2 |ψtt |2dxdy − K (ψt + wxt ) ψt dxdy − d1 ψtt ψt dxdy
Ω
Ω
Ω
Ω
Ω
Ω
+ ρ2
|ϕtt |2dxdy − K (ϕt + wxt ) ϕt dxdy − d2 ϕtt ϕt dxdy.
Then, defining
(4.13)
F4 (t) :=
[ρ2 ψtt ψt + ρ2 ϕtt ϕt + K∇w.(ψt , ϕt )] dxdy
(4.14)
Ω
and using (4.11) we obtain
d
F4 (t) ≤ −Dα ||ψt ||2H 1 + ||ϕt ||2H 1 + C
dt
+K (wx ψtt + wy ϕtt ) dxdy.
|ψtt |2 + |ϕtt |2 dxdy
Ω
(4.15)
Ω
Let
F5 (t) :=
F3 (t) +
C
F4 (t).
Dα
(4.16)
Then, from (4.10) and (4.15) we have
d
CK
F5 (t) ≤ −2E1 (t)+C
|ψtt |2 +|ϕtt |2 +|ψttt |2 dxdy+
(wx ψtt +wy ϕtt ) dxdy
dt
Dα Ω
Ω
1−µ
4ρ1 D
(4.17)
−
ψxtt +µϕytt ,
(ψytt −ϕxtt ) .∇wq(x, y) dxdy.
K
2
Ω
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ON THE STABILITY OF MINDLIN–TIMOSHENKO PLATES
261
Note that, using the definition of E1 (t) and the inequality (4.12), we obtain
β
||∇w||2L2 .
2
Therefore, applying (4.18) in (4.17) we can deduce that
d
β
F5 (t) ≤ −E1 (t) − ||∇w||2L2 + C |ψttt |2 dxdy
dt
4
Ω
2
2
+Cβ ||ψtt ||H 1 + ||ϕtt ||H 1 ,
− 2E1 (t) ≤ −E1 (t) −
(4.18)
(4.19)
where Cβ > 0 is defined as in (4.7) and depends also on β > 0.
Similarly as in (4.13), differentiating equations (1.12)–(1.13) with respect to t two
times, and multiplying by ψtt and ϕtt respectively, we can deduce
d
ρ2 [ψttt ψtt + ϕttt ϕtt ] dxdy
dt Ω
≤ −Dα ||ψtt ||2H 1 + ||ϕtt ||2H 1 + ρ2 |ψttt |2 + |ϕttt |2 dxdy
Ω
−K
−K
(ψtt + wxtt ) ψtt dxdy − d1
Ω
ψttt ψtt dxdy
Ω
(ϕtt + wxtt ) ϕtt dxdy − d2
Ω
ϕttt ϕtt dxdy,
(4.20)
Ω
where inequality (4.11) is used. We define
F6 (t) := [ρ2 ψttt ψtt + ρ2 ϕttt ϕtt − K∇wt .(ψtt , ϕtt ) + K∇w.(ψttt , ϕttt )] dxdy. (4.21)
Ω
Then, from (4.20) we deduce
d
|ψttt |2 + |ϕttt |2 dxdy
F6 (t) ≤ −Dα ||ψtt ||2H 1 + ||ϕtt ||2H 1 + C
dt
Ω
−K ∇w.(ψtttt , ϕtttt ) dxdy.
(4.22)
Ω
Finally we define
F7 (t) :=
F5 (t) +
Cβ
F6 (t).
Dα
Then, from (4.19) and (4.22) we obtain
d
F7 (t) ≤
dt
−E1 (t) −
−K
β
||∇w||2L2 + C
4
(4.23)
|ψttt |2 + |ϕttt |2 dxdy
Ω
∇w.(ψtttt , ϕtttt ) dxdy,
Ω
and we can deduce that
d
|ψttt |2 + |ϕttt |2 + |ψtttt |2 + |ϕtttt |2 dxdy,
F7 (t) ≤ −E1 (t) + C0
dt
Ω
(4.24)
where C0 > 0 is a constant defined as in (4.7), and also depends on the constants given
by Korn’s Inequality.
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262
HUGO D. FERNÁNDEZ SARE
Now we are in the position to prove the main result of this section.
Theorem 4.2. Suppose that the initial data verify
U0 := (w0 , w1 , ψ0 , ψ1 , ϕ0 , ϕ1 ) ∈ D(A4 ).
Then the first order energy E1 (t) associated to the system (1.11)–(1.13) with boundary
conditions (2.1) decays polynomially to zero as time goes to infinity; that is, there exists
a positive constant C, being independent of the initial data, such that
E1 (t) ≤
4
C
Ei (0).
t i=1
(4.25)
Moreover, if U0 ∈ D(A4k ), then
Ck
||A4k U0 ||H , ∀k = 1, 2, 3, ...,
(4.26)
tk
where {T (t)}t≥0 is the semigroup associated to the system (1.11)–(1.13) with infinitesimal
generator A defined as in (2.3).
||T (t)U0 ||H ≤
Proof. We define L(t) as
L(t) :=
4
C0 Ei (t) + F7 (t),
d i=1
where d := min{d1 , d2 } > 0, with d1 , d2 given by the system (1.11)–(1.13). Then, using
(4.3) and (4.24) we obtain
d
L(t) ≤ −E1 (t).
dt
Therefore
t
E1 (s)ds ≤ L(0) − L(t), ∀t ≥ 0.
(4.27)
0
On the other hand, it is not difficult to prove that there exists a constant C > 0 such
that
4
Ei (0), ∀t ≥ 0.
(4.28)
L(0) − L(t) ≤ C
i=1
From (4.27)–(4.28) we obtain
t
E1 (s)ds ≤ C
0
Then, since
4
Ei (0).
(4.29)
i=1
d
d
tE1 (t) = E1 (t) + t E1 (t) ≤ E1 (t),
dt
dt
from (4.29) we get
E1 (t) ≤
4
C
Ei (0),
t i=1
which completes (4.25) and shows that (4.26) holds, for k = 1.
Finally, if U0 ∈ D(A4k ) k ≥ 2, using that 0 ∈ ρ(A) (resolvent set of A), we can apply
[13, Proposition 3.1] to obtain (4.26), which completes the proof.
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ON THE STABILITY OF MINDLIN–TIMOSHENKO PLATES
263
Acknowledgements. The author should like to thank Professor Dr. R. Racke for
his suggestions, corrections, and comments on this work.
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