On the stability of Mindlin-Timoshenko plates

Universität Konstanz

On the stability of Mindlin-Timoshenko plates

Hugo D. Fernández Sare

Konstanzer Schriften in Mathematik und InformatikNr. 237, Oktober 2007

ISSN 1430-3558

© Fachbereich Mathematik und Statistik© Fachbereich Informatik und InformationswissenschaftUniversität KonstanzFach D 188, 78457 Konstanz, GermanyE-Mail: [email protected]: http://www.informatik.uni-konstanz.de/Schriften/

mailto:[email protected]

http://www.ub.uni-konstanz.de/kops/volltexte/2007/3826/

http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-38264

ON THE STABILITY OF MINDLIN–TIMOSHENKO PLATES

HUGO D. FERNANDEZ SARE

Abstract. We consider a Mindlin-Timoshenko model with frictional dissipations actingon the equations for the rotation angles. We prove that this system is not exponentiallystable independent of any relations between the constants of the system, which is differentfrom the analogous’ one-dimensional case. Moreover, we show that the solution decayspolynomially to zero, with rates that can be improved depending on the regularity of theinitial data.

1. Introduction

The conservative Mindlin-Timoshenko model in two dimensional case is given by

ρhwtt −K

[∂

∂x

(ψ +

∂w

∂x

)+

∂

∂y

(ϕ+

∂w

∂y

)]= 0 (1.1)

ρh3

12ψtt −D

(∂2ψ

∂x2+

1− µ

2∂2ψ

∂y2+

1 + µ

2∂2ϕ

∂x∂y

)+K

(ψ +

∂w

∂x

)= 0 (1.2)

ρh3

12ϕtt −D

(∂2ϕ

∂y2+

1− µ

2∂2ϕ

∂x2+

1 + µ

2∂2ψ

∂x∂y

)+K

(ϕ+

∂w

∂y

)= 0 (1.3)

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 < 1

2 in physical situations), D is the modulusof 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, ϕare 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 thathere another equation for rotation angles is considered. Note also that the coupling betweenthe equations of the rotational angles (ψ,ϕ) and the displacement equation w, is weakerthan in one dimension. Therefore the questions how it is possible to stabilize the systemand to find ”sufficient” dissipations to produce exponential stability are interesting and thusis not much studied in the literature.

In [7], Lagnese considered a bounded domain Ω having a Lipschitz boundary Γ such thatΓ = Γ0 ∪ Γ1, where Γ0 and Γ1 are relatively open, disjoints subsets of Γ with Γ1 6= ∅. Heconsidered the following boundary conditions

2000 Mathematics Subject Classification. 35 B 40, 74 H 40.Key words and phrases. Timoshenko plates, non-exponential stability, polynomial stabilitiy.The author was supported by the DFG-project “Hyperbolic Thermoelasticity” (RA 504/3-3).

1

2 HUGO D. FERNANDEZ SARE

w = ψ = ϕ = 0 in Γ0 (1.4)

K

(∂w

∂x+ ψ,

∂w

∂y+ ϕ

)· ν = m1 in Γ1 (1.5)

D

(∂ψ

∂x+ µ

∂ϕ

∂y,

1− µ

2

(∂ϕ

∂x+∂ψ

∂y

))· ν = m2 in Γ1 (1.6)

D

(1− µ

2

(∂ϕ

∂x+∂ψ

∂y

),∂ϕ

∂y+ µ

∂ψ

∂x

)· ν = m3 in Γ1, (1.7)

where ν := (ν1, ν2) is the unit exterior normal to Γ = ∂Ω and m1,m2,m3 are the linearboundary feedbacks given by

m1,m2,m3 = −Fwt, ψt, ϕtwith F = [fij ] a 3 × 3 matrix of real L∞(Γ1) functions such that F is symmetric andpositive semidefinite on Γ1. In that conditions, Lagnese proved that the system (1.1)–(1.3)is exponentially stable, without any restrictions on the coefficients of the system. The sameresult is obtained by M. Rivera and P. Oquendo [10], where they consider in (1.5)–(1.7)boundary dissipations of memory type, that is

w +K

∫ t

0

g1(t− s)(∂w

∂x(s) + ψ(s),

∂w

∂y(s) + ϕ(s)

)· νds = 0 in Γ1

ψ +D

∫ t

0

g2(t− s)(∂ψ(s)∂x

+ µ∂ϕ(s)∂y

,1− µ

2

(∂ϕ(s)∂x

+∂ψ(s)∂y

))· νds = 0 in Γ1

ϕ+D

∫ t

0

g3(t− s)(

1− µ

2

(∂ϕ(s)∂x

+∂ψ(s)∂y

),∂ϕ(s)∂y

+ µ∂ψ(s)∂x

)· νds = 0 in Γ1.

With these boundary feedbacks, together with condition (1.4), they proved that the solutionsof the system (1.1)–(1.3) are exponentially stable provided that the kernels have exponentialbehavior, and are polynomially stable for kernels of polynomial type. Similar dissipationsare used by L. 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 internaldissipations into the system, this 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 naturalquestions are the following: what happens if we remove one of these dissipations? and, ofcourse, which is the ”natural” candidate to be removed?. Looking to the one dimensionalmodel we can deduce some conclusions in order to solve these questions. In [12], M. Riveraand Racke considered the one dimensional damped Timoshenko system

ρ1ϕtt − k(ϕx + ψ)x = 0 in (0, L) (1.8)ρ2ψtt − 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 wavespeeds of the system are equal, that is if

ρ1

k=ρ2

b. (1.10)

That Timoshenko model (1.8)-(1.9) with several type of dissipations has been studied bymany authors, see for example [4, 5, 6, 11, 12, 14] and the references therein. The common

ON THE STABILITY OF MINDLIN–TIMOSHENKO PLATES 3

point in almost every work is the condition (1.10), necessary and sufficient to obtain ex-ponential 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 equiva-lent problem in two dimensions. This is, introducing frictional dissipations in the rotationalangle equations (1.2)-(1.3), we obtain a new dissipative system where the rate of decay forthe 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 Timoshenkosystem

ρ1wtt −Kdiv(ψ +

∂w

∂x, ϕ+

∂w

∂y

)= 0 (1.11)

ρ2ψtt −Ddiv(∂ψ

∂x+ µ

∂ϕ

∂y,1− µ

2

(∂ϕ

∂x+∂ψ

∂y

))+K

[ψ +

∂w

∂x

]+ d1ψt = 0 (1.12)

ρ2ϕtt −Ddiv(

1− µ

2

(∂ϕ

∂x+∂ψ

∂y

),∂ϕ

∂y+ µ

∂ψ

∂x

)+K

[ϕ+

∂w

∂y

]+ d2ϕt = 0 (1.13)

where ρ1 := ρh, and ρ2 := ρh3

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 constantsof the system, which is a different result from the one obtained in [12] in the one-dimensionalcase, where the condition (1.10) was sufficient and necessary to obtain exponential stability.Moreover, using multiplier techniques and Pruss’ result [13], we will prove that the system(1.11)–(1.13) is polynomially stable with rates that can be improved depending on the initialdata. We would like to add here that this is the first time that the asymptotic behaviorof the system (1.11)–(1.13) is studied, and that our analysis shows the differences in thedimensions clearly.

The paper is organized as follows: In Section 2 we shall look to the existence and unique-ness results using semigroup theory. The Timoshenko system (1.11)–(1.13) is shown to benot exponentially stable subject to mixed boundary conditions in Section 3. Finally, in Sec-tion 4 we study the polynomial stability of the system (1.11)–(1.13) with Dirichlet boundaryconditions.

2. Existence and uniqueness

We will use the standard notation Hk(Ω) or Hk0 (Ω) to denote usual Sobolev spaces of

order k over the regular domain Ω, and set L2(Ω) = H0(Ω). We consider the Timoshenkosystem (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 (2.2)

ϕ(x, 0) = ϕ0(x) , ϕt(x, 0) = ϕ1(x) , in Ω.

In order to obtain existence, uniqueness and stability results we will use semigroup theory.For this purpose we rewrite the system as evolution equation for

U = (w,wt, ψ, ψt, ϕ, ϕt)′ ≡ (u1, u2, u3, u4, u5, u6, )′.


Then U formally satisfiesUt = A1U, U(0) = U0

where U0 = (w0, w1, ψ0, ψ1, ϕ0, ϕ1)′, and A1 is the (yet formal) differential operator

A1 :=

0 Id 0 0 0 0

Kρ1

∆ 0 Kρ1∂x 0 K

ρ1∂y 0

0 0 0 Id 0 0

−Kρ2∂x 0 B1 −d1

ρ2Id D

ρ2

(1+µ

2

)∂2

xy 0

0 0 0 0 0 Id

−Kρ2∂y 0 D

ρ2

(1+µ

2

)∂2

xy 0 B2 −d2ρ2Id

, (2.3)

where the differential operators Bi (i = 1, 2), are defined by

B1 =D

ρ2

[∂2

x +(

1− µ

2

)∂2

y

]− k

ρ2Id

B2 =D

ρ2

[(1− µ

2

)∂2

x + ∂2y

]− k

ρ2Id.

LetH1 := H1

0 (Ω)× L2(Ω)×H10 (Ω)× L2(Ω)×H1

0 (Ω)× L2(Ω)be the Hilbert space. In order to endow the space H1 with a norm associated to the energyof 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) ∈ [H10 (Ω)]2,

∫

Ω

[|ψx|2 + |ϕy|2 +

(1− µ

2

)|ψy + ϕx|2 + µψxϕy + µϕyψx

]dxdy ≥ α

[||ψ||2H1 + ||ϕ||2H1

].

Moreover, for every K0 > 0 there exists β := β(K0) > 0 such that for all K ≥ K0 and forall (w,ψ, ϕ) ∈ [H1

0 (Ω)]3,∫

Ω

[|ψx|2 + |ϕy|2 +

(1− µ

2

)|ψy + ϕx|2 + µψxϕy + µϕyψx

]dxdy

+K∫

Ω

|ψ + wx|2 dxdy +K

∫

Ω

|ϕ+ wx|2 dxdy ≥ β[||∇ψ||2L2 + ||∇ϕ||2L2 + ||∇w||2L2

].

Proof. It’s 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||u5

y||2L2

+K||u3 + u1x||2L2 +K||u5 + u1

y||2L2 + µ(u3x, u

5y)L2 + µ(u5

y, 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 thefollowing result about existence and uniqueness of solutions.


Theorem 2.2. Let U0 = (w0, w1, ψ0, ψ1, ϕ0, ϕ1)′ ∈ H1. Then there exists a unique solutionU(t) = (w,wt, ψ, ψt, ϕ, ϕt)′ to system (1.11)–(1.13) with Dirichlet boundary conditions (2.1)satisfying

U ∈ C(R+;D(A1)) ∩ C1(R+;H1).

Moreover, if U0 ∈ D(An1 ), then

U ∈ Cn−k(R+;D(Ak1)) , k = 0, 1, · · ·, n.

Remark 2.3. The same analysis can be applied to obtain existence and uniqueness resultsfor mixed boundary conditions.

3. Non-exponential stability

In this Section we will prove that system (1.11)-(1.13) is not exponentially stable forsuitable 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 :=

(x, y) : 0 < x < L1 , y = 0, L2

Γ2 :=

(x, y) : 0 < y < L2 , x = 0, L1

.

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 = 0 ,

(1− µ

2

(∂ϕ

∂x+∂ψ

∂y

),∂ϕ

∂y+ µ

∂ψ

∂x

)· ν = 0 in Γ1 (3.1)

ϕ = 0 ,

(∂ψ

∂x+ µ

∂ϕ

∂y,

1− µ

2

(∂ϕ

∂x+∂ψ

∂y

))· ν = 0 in Γ2

where ν := (ν1, ν2) is the unit exterior normal to Γ = ∂Ω. Therefore, the semigroupformulation is given in the Hilbert space,

H2 := H10 (Ω)× L2(Ω)×H1

Γ1(Ω)× L2(Ω)×H1

Γ2(Ω)× L2(Ω),

whereH1

Γi(Ω) :=

u ∈ H1(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 etAt≥0 in a Hilbert space with norm ‖ · ‖ isexponentially stable if and only if

(i) the resolvent set %(A) of A contains the imaginary axis

and

(ii) lim supλ→±∞

‖(iλId−A)−1‖ <∞

hold.


Hence it suffices to show the existence of sequences (λn)n ⊂ R with

limn→∞

|λn| = ∞,

and (Un)n ⊂ D(A1), (Fn)n ⊂ H, such that

(iλnId−A1)Un = Fn is bounded and limn→∞

‖Un‖H1 = ∞.

As Fn ≡ F we choose F := (0, f2, 0, f4, 0, f6)′ with

f2 := F 2 sin(δλ1x) sin(δλ2y) , F 2 6= 0 (constant)

f4 := F 4 cos(δλ1x) sin(δλ2y) , F 4 6= 0 (constant)

f6 := F 6 sin(δλ1x) cos(δλ2y) , F 6 6= 0 (constant),

where

λ1 ≡ λ1,n :=nπ

δL1, λ2 ≡ λ2,n :=

nπ

δL2(n ∈ N) , δ :=

√ρ1

k.

Finally we deffine

λ ≡ λn :=√λ2

1 + λ22. (3.2)

The solution U = (v1, v2, v3, v4, v5, v6)′ of the resolvent equation

(iλId−A1)U = F

should satisfy

iλv1 − v2 = 0

iλv2 − k

ρ1(v3 + v1

x)x −− k

ρ1(v5 + v1

y)y = f2

iλv3 − v4 = 0

iλv4 − D

ρ2

[v3

xx +(

1− µ

2

)v3

yy +(

1 + µ

2

)v5

xy

]+

k

ρ2(v3 + v1

x) +d1

ρ2v4 = f4

iλv5 − v6 = 0

iλv6 − D

ρ2

[(1− µ

2

)v5

xx + v5yy +

(1 + µ

2

)v3

xy

]+

k

ρ2(v5 + v1

y) +d2

ρ2v6 = f6.

(3.3)

Eliminating v2, v4, v6 we obtain for v1, v3, v5 the following system

−λ2ρ1v1 − k(v3 + v1

x)x − k(v5 + v1y)y = ρ1f

2

−λ2ρ2v3 −D

[v3

xx +(

1−µ2

)v3

yy +(

1+µ2

)v5

xy

]+ k(v3 + v1

x) + iλd1v3 = ρ2f

4

−λ2ρ2v5 −D

[(1−µ

2

)v5

xx + v5yy +

(1+µ

2

)v3

xy

]+ k(v5 + v1

y) + iλd2v5 = ρ2f

6.

(3.4)

System (3.4) can be solved by

v1(x, y) := A sin(δλ1x) sin(δλ2y)v3(x, y) := B cos(δλ1x) sin(δλ2y)v5(x, y) := C sin(δλ1x) cos(δλ2y)

where A, B, C depend on λ and will be determined explicitly in the sequel. Note that thischoice is just compatible with the boundary conditions (3.1). System (3.4) is equivalent to


finding A,B,C such that

−λ2ρ1A+ kδ2(λ2

1 + λ22

)A+ kδλ1B + kδλ2C = ρ1F

2 (3.5)

−λ2ρ2B +Dδ2λ21B +D

(1− µ

2

)δ2λ2

2B +D

(1 + µ

2

)δ2λ1λ2C+

+kB + kδλ1A+ iλd1B = ρ2F4 (3.6)

−λ2ρ2C +D

(1− µ

2

)δ2λ2

1C +Dδ2λ22C +D

(1 + µ

2

)δ2λ1λ2B+

+kC + kδλ2A+ iλd2C = ρ2F6. (3.7)

Using the definitions of δ and λ, we obtain from (3.5) that

B = −λ2

λ1C +

1λ1δF 2 (3.8)

or

C = −λ1

λ2B +

1λ2δF 2. (3.9)

Using (3.9) in (3.6) results[−λ2

(ρ2 −Dδ2

(1− µ

2

))+ k + iλd1

]B + kδλ1A = ρ2F

4 −D

(1 + µ

2

)δ3λ1F

2. (3.10)

Similarly, using (3.8) in (3.7) results[−λ2

(ρ2 −Dδ2

(1− µ

2

))+ k + iλd2

]C + kδλ2A = ρ2F

6 −D

(1 + µ

2

)δ3λ2F

2. (3.11)

Let

Θ := ρ2 −Dρ1

k

(1− µ

2

), (3.12)

then, using the definition of δ, we obtain from (3.10)-(3.11) that A,B satisfies

(−λ2Θ + k + iλd1

)B + kδλ1A = ρ2F

4 −D

(1 + µ

2

)δ3λ1F

2 (3.13)

(−λ2Θ + k + iλd2

)C + kδλ2A = ρ2F

6 −D

(1 + µ

2

)δ3λ2F

2. (3.14)

Remark 3.2. Note that the condition Θ = 0 in (3.12) gives a relationship (in 2-dimensionalcase) similar to the relation

ρ1

k=ρ2

b,

which is necessary and sufficient condition for exponential stability in 1-dimensional case,see [12]. We will show that the system (1.11)-(1.13) is non-exponentially stable, independentof any relation between the coefficients of the system, in particular of Θ = 0 in (3.12).

Using (3.9) into (3.14) results

(−λ2Θ + k + iλd2

) λ21

λ22

B − kδλ1A = −λ1

λ2ρ2F

6 +D

(1 + µ

2

)δ3λ1F

2

+(−λ2Θ + k + iλd2

)δλ1F

2, (3.15)


then, adding the equalities (3.13) and (3.15) yields[(−λ2Θ + k + iλd1

)+

(−λ2Θ + k + iλd2

) λ21

λ22

]B = ρ2F

4 − λ1

λ2ρ2F

6

+(−λ2Θ + k + iλd2

)δλ1F

2,

this is

B =ρ2F

4 − λ1

λ2ρ2F

6 +(−λ2Θ + k + iλd2

)δλ1F

2

−λ2Θ(

1 +λ2

1

λ22

)+ k

(1 +

λ21

λ22

)+ iλ

(d1 + d2

λ21

λ22

) (3.16)

and using (3.13) we have

A =ρ2

kδλ1F 4 −D

(1 + µ

2

)δ2

kF 2 − (−λ2Θ + k + iλd1

)B (3.17)

with B given by (3.16). We define

Q(λ) := −λ2Θ(

1 +λ2

1

λ22

)+ k

(1 +

λ21

λ22

)+ iλ

(d1 + d2

λ21

λ22

),

where, using the definitions of λi(i = 1, 2), we can conclude that

L :=λ1

λ2=

L2

L1> 0. (3.18)

Therefore Q(λ) is given by

Q(λ) = −λ2Θ(1 + L2

)+ k

(1 + L2

)+ iλ

(d1 + d2L

2). (3.19)

We also define the following functions

A1(λ) :=ρ2

kδλ1F 4 +

1Q(λ)

[kρ2

(LF 6 − F 4

)−D

(1 + µ

2

)δ2

(1 + L2

)](3.20)

A2(λ) :=1

Q(λ)

[− λ4λ1Θ2δF 2 + iλ3λ1Θ(d1 − d2) δF 2 + λ2λ1 (d2d1 + 2Θk) δF 2

+λ2Θ((F 4 − LF 6)ρ2 +D

(1 + µ

2

)δ2

k(1 + L2)

)− iλλ1 (d1 + d2) kδF 2

+iλ(d1ρ2(LF 6 − F 4)−D

(1 + µ

2

)(d1 + d2L

2))− δλ1kF

2

]. (3.21)

Then we have in (3.17) thatA = A1(λ) +A2(λ).

Recalling thatv2 = iλv1 = iλA sin(δλ1x) sin(δλ2y)

we get

v2 =(iλA1(λ) + iλA2(λ)

)sin(δλ1x) sin(δλ2y).


Note that

||Un||H ≥ ||v2||L2

=( ∫

Ω

|v2|2dxdy)1/2

≥ −C1|λA1(λ)|+ C2|λA2(λ)| ,where Ci := Ci(L1, L2) > 0, i = 1, 2. Then, to complete our result, it is sufficient to showthat,

(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.In fact, using the definitions of λ, λ1 in (3.20) we obtain

λA1(λ) =ρ2

kδ

√1 +

1L2F 4 +

kρ2

(LF 6 − F 4

)−D

(1 + µ

2

)δ2

(1 + L2

)

−λΘ(1 + L2

)+k

λ

(1 + L2

)+ i

(d1 + d2L

2) .

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 6= 0. When Θ = 0 we have in (3.21) that

λA2(λ) =1

Q0(λ)

[2λ3λ1d2d1δF

2 − iλ2λ1 (d1 + d2) kδF 2

+iλ2(d1ρ2(LF 6 − F 4)−D

(1 + µ

2

)(d1 + d2L

2))− δλλ1kF

2

],

withQ0(λ) = k(1 + L2) + iλ(d1 + d2L

2).Therefore |λA2(λ)| −→ ∞. Thus we have proved

Theorem 3.3. The Timoshenko system (1.11)-(1.13) with boundary conditions (3.1) is notexponentially stable, independent of any relation between the constants of the system.

Remark 3.4. As in the 1-dimensional case, the non-exponential stability to Dirichlet bound-ary conditions (2.1) is still an open problem. Note also that the function that generates thenon-exponential stability, that is λA2(λ)λ, has the behavior as |λA2(λ)| ∼ (λ3), whichproduce the expectation that, to show polynomial stability results, we will need energies ofhigher 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)=12

∫

Ω

[ρ1|wt|2 + ρ2|ψt|2 + ρ2|ϕt|2 +K|ψ + wx|2 +K|ϕ+ wx|2 +D|ψx|2

+D|ϕy|2 +(

1− µ

2

)D|ψy + ϕx|2 + 2Dµψxϕy

]dxdy, (4.1)


which is obtained multiplying the equations (1.11) by wt, (1.12) by ψt and (1.13) by ϕt.Also, we can define the energies

Ei+1(t) := E1(t; ∂(i)t w, ∂

(i)t ψ, ∂

(i)t ϕ) , i = 1, 2, 3. (4.2)

It is not difficult to show thatd

dtEi(t) = −d1

∫

Ω

|∂(i)t ψ|2dxdy − d2

∫

Ω

|∂(i)t ϕ|2dxdy , i = 1, 2, 3, 4. (4.3)

We define

F1(t) :=∫

Ω

[ρ1wtw + ρ2ψtψ + ρ2ϕtϕ+d1

2ψ2 +

d2

2ϕ2] dxdy, (4.4)

then, multiplying the equation (1.11) by w, (1.12) by ψ and (1.13) by ϕ, results in

d

dtF1(t) = −D

∫

Ω

[|ψx|2 + |ϕy|2 +

(1− µ

2

)|ψy + ϕx|2 + 2µψyϕx

]dxdy

−K∫

Ω

|ψ + wx|2 dxdy −K

∫

Ω

|ϕ+ wx|2 dxdy + ρ2

∫

Ω

|ψt|2 dxdy

+ρ2

∫

Ω

|ϕt|2 dxdy + ρ1

∫

Ω

|wt|2 dxdy. (4.5)

Let q : Ω → R defined by q(x, y) = x. We define

F2(t) := −D∫

Ω

(ψxt + µϕyt,

(1− µ

2

)(ψyt − ϕxt)

).∇wq(x, y) dxdy, (4.6)

then, differentiating the equation (1.12) with respect to t and multiplying by q(x, y)wt inL2(Ω) results in

d

dtF2(t) = −K

∫

Ω

|wt|2dxdy + ρ2

∫

Ω

ψtttq(x, y)wt dxdy +K

∫

Ω

ψtq(x, y)wt dxdy

+d1

∫

Ω

ψttq(x, y)wt dxdy +D

∫

Ω

(ψxt + µϕyt)wt dxdy

−D∫

Ω

(ψxtt + µϕytt,

(1− µ

2

)(ψytt − ϕxtt)

).∇wq(x, y) dxdy,

where we can conclude that there exists

Ci := Ci(ρ1, ρ2,K,D, µ,Ω) > 0 , i = 1, 2, (4.7)

such thatd

dtF2(t) ≤ −K

2

∫

Ω

|wt|2dxdy + C1

[||ψt||2H1 + ||ϕt||2H1

]+ C2

∫

Ω

[|ψtt|2 + |ψttt|2]dxdy

−D∫

Ω

(ψxtt + µϕytt,

(1− µ

2

)(ψytt − ϕxtt)

).∇wq(x, y) dxdy. (4.8)

We will use the letter C to denote several positive constants defined as in (4.7). Defining

F3(t) := F1(t) +4ρ1

KF2(t), (4.9)


and using (4.5) and (4.8) we have

d

dtF3(t) ≤ −2E1(t) + C

[||ψt||2H1 + ||ϕt||2H1

]+ C

∫

Ω

[|ψtt|2 + |ψttt|2]dxdy

−4ρ1D

K

∫

Ω

(ψxtt + µϕytt,

(1− µ

2

)(ψytt − ϕxtt)

).∇wq(x, y) dxdy.(4.10)

Remark 4.1. In (4.10) we have already the first order energy with negative sign, but it isnecessary to estimate other higher-order terms. The following functionals will be defined inorder to estimate these terms.

First, note that using Korn’s Inequality [2], we have that there exists constants α, β > 0such that (see [7])

∫

Ω

[|ψx|2 + |ϕy|2 +

(1− µ

2

)|ψy + ϕx|2 + 2µψxϕy

]dxdy ≥ α

[||ψ||2H1 + ||ϕ||2H1

](4.11)

and∫

Ω

[|ψx|2 + |ϕy|2 +

(1− µ

2

)|ψy + ϕx|2 + 2µψxϕy

]dxdy

+K∫

Ω

|ψ + wx|2 dxdy +K

∫

Ω

|ϕ+ wx|2 dxdy ≥ β[||∇ψ||2L2 + ||∇ϕ||2L2 + ||∇w||2L2

]. (4.12)

On the other hand, differentiating the equations (1.12)-(1.13) with respect to t and multi-plying by ψt and ϕt respectively, results in

d

dt

∫

Ω

ρ2 [ψttψt + ϕttϕt]dxdy = −D∫

Ω

[|ψxt|2+|ϕyt|2+

(1− µ

2

)|ψyt+ϕxt|2+2µψytϕxt

]dxdy

+ρ2

∫

Ω

|ψtt|2dxdy −K

∫

Ω

(ψt + wxt)ψt dxdy − d1

∫

Ω

ψttψt dxdy

+ρ2

∫

Ω

|ϕtt|2dxdy −K

∫

Ω

(ϕt + wxt)ϕt dxdy − d2

∫

Ω

ϕttϕt dxdy. (4.13)

Then, defining

F4(t) :=∫

Ω

[ρ2ψttψt + ρ2ϕttϕt +K∇w.(ψt, ϕt)] dxdy (4.14)

and using (4.11) we obtain

d

dtF4(t) ≤ −Dα [||ψt||2H1 + ||ϕt||2H1

]+ C

∫

Ω

[|ψtt|2 + |ϕtt|2]dxdy

+K∫

Ω

(wxψtt + wyϕtt) dxdy. (4.15)

Let

F5(t) := F3(t) +C

DαF4(t), (4.16)


then, from (4.10) and (4.15) we haved

dtF5(t) ≤ −2E1(t) + C

∫

Ω

[|ψtt|2 + |ϕtt|2 + |ψttt|2]dxdy +

CK

Dα

∫

Ω

(wxψtt + wyϕtt) dxdy

−4ρ1D

K

∫

Ω

(ψxtt + µϕytt,

(1− µ

2

)(ψytt − ϕxtt)

).∇wq(x, y) dxdy. (4.17)

Note that, using the definition of E1(t) and the inequality (4.12), we obtain

−2E1(t) ≤ −E1(t)− β

2||∇w||2L2 . (4.18)

Therefore, applying (4.18) in (4.17) we can deduce thatd

dtF5(t) ≤ −E1(t)− β

4||∇w||2L2 + C

∫

Ω

|ψttt|2 dxdy

+Cβ

[||ψtt||2H1 + ||ϕtt||2H1

], (4.19)

where Cβ > 0 is defined as (4.7) and depends also of 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 deduced

dt

∫

Ω

ρ2 [ψtttψtt + ϕtttϕtt]dxdy ≤ −Dα [||ψtt||2H1 + ||ϕtt||2H1

]+ ρ2

∫

Ω

[|ψttt|2 + |ϕttt|2]dxdy

−K∫

Ω

(ψtt + wxtt)ψtt dxdy − d1

∫

Ω

ψtttψtt dxdy

−K∫

Ω

(ϕtt + wxtt)ϕtt dxdy − d2

∫

Ω

ϕtttϕtt dxdy, (4.20)

where inequality (4.11) is used. We define

F6(t) :=∫

Ω

[ρ2ψtttψtt + ρ2ϕtttϕtt −K∇wt.(ψtt, ϕtt) +K∇w.(ψttt, ϕttt)] dxdy. (4.21)

Then, from (4.20) we deduced

dtF6(t) ≤ −Dα [||ψtt||2H1 + ||ϕtt||2H1

]+ C

∫

Ω


−K∫

Ω

∇w.(ψtttt, ϕtttt) dxdy (4.22)

Finally we define

F7(t) := F5(t) +Cβ

DαF6(t). (4.23)

Then, from (4.19) and (4.22) results ind

dtF7(t) ≤ −E1(t)− β

4||∇w||2L2 + C

∫

Ω


−K∫

Ω

∇w.(ψtttt, ϕtttt) dxdy,

and we can deduce thatd

dtF7(t) ≤ −E1(t) + C0

∫

Ω

[|ψttt|2 + |ϕttt|2 + |ψtttt|2 + |ϕtttt|2]dxdy, (4.24)


where C0 > 0 is a constant defined as (4.7), and also depends on the constants given byKorn’s Inequality.

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 system (1.11)-(1.13) with boundary conditions(2.1) decays polynomially to zero as time goes to infinity, that is, there exists a positiveconstant C, being independent of the initial data, such that

E1(t) ≤ C

t

4∑

i=1

Ei(0). (4.25)

Moreover, if U0 ∈ D(A4k), then

||T (t)U0||H ≤ Ck

tk||A4kU0||H , ∀k = 1, 2, 3, ... (4.26)

where T (t)t≥0 is the semigroup associated to system (1.11)-(1.13) with infinitesimal gen-erator A defined as (2.3).

Proof. We define L(t) as

L(t) :=C0

d

4∑

i=1

Ei(t) + F7(t),

where d := mind1, d2 > 0, with d1, d2 given by the system (1.11)-(1.13). Then, using (4.3)and (4.24) we obtain

d

dtL(t) ≤ −E1(t).

Therefore ∫ t

0

E1(s)ds ≤ L(0)− L(t) , ∀t ≥ 0. (4.27)

On the other hand, it is not difficult to prove that there exists a constant C > 0 such that

L(0)− L(t) ≤ C

4∑

i=1

Ei(0) , ∀t ≥ 0. (4.28)

From (4.27)–(4.28) we obtain∫ t

0

E1(s)ds ≤ C

4∑

i=1

Ei(0). (4.29)

Then, sinced

dt

tE1(t)

= E1(t) + t

d

dtE1(t) ≤ E1(t),

from (4.29) we get

E1(t) ≤ C

t

4∑

i=1

Ei(0),

which completes (4.25) and show that (4.26) holds, for k = 1.Finally, if U0 ∈ D(A4k), k ≥ 2, we use Pruss’ results [13] to obtain (4.26), which completes

the proof. ¤


References

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Chapman&Hall/CRC, Boca Raton (1999).10. Munoz Rivera, J.E., Portillo Oquendo, H.: Asymptotic behavior on a Mindlin–Timoshenko plate with

viscoelastic dissipation on the boundary. Funkcialaj Ekvacioj 46 (2003), 363–382.11. Munoz Rivera, J.E., Racke, R.: Mildly dissipative nonlinear Timoshenko systems — global existence

and exponential stability. J. Math. Anal. Appl. 276 (2002), 248–278.12. Munoz Rivera, J.E., Racke, R.: Global stability for damped Timoshenko systems. Disc. Cont. Dyn.

Sys. 9 (2003), 1625–1639.13. Pruss, J., Batkai, A., Engel, K., Schnaubelt, R.: Polynomial stability of operator semigroups. Math.

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731–734.

Department of Mathematics and Statistics, University of Konstanz, 78457, Konstanz, GermanyE-mail address: [email protected]

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On the stability of Mindlin-Timoshenko plates