On the asymptotics of solutions to resonator equations

Universität Konstanz

On the asymptotics of solutions to resonator equations

Bugra Kabil

Konstanzer Schriften in Mathematik Nr. 281, August 2011

ISSN 1430-3558

© Fachbereich Mathematik und Statistik

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-144793

On the asymptotics of solutions to resonator equations

BUĞRA KABIL

Department of Mathematics and Statistics University of Konstanz

78457 Konstanz, Germany bugra.kabil@uni-konstanz.de

Abstract

In this paper, we consider a system of micro-beam resonators within the thermoelastic theory of Lord and Shulman. It is a particular case of a thermoelastic system given by a coupling of a plate equation to a hyperbolic heat equation arising from Cattaneo’s law of heat conduction. In a bounded domain, the system can be damped by a term such that it is exponentially stable. In the whole space, one can determine the decay rates for the system.

Keywords: hyperbolic models in thermoelasticity, mechanical resonator, exponential stability, decay rates.

1 Introduction

Resonators are systems which naturally oscillate at some frequencies, called its resonant frequencies. There are many kinds of resonators. We consider mechanical resonators.

Microresonators have high sensitivity at room temperature. Thermoelastic damping is one of the reasons for the dissipation or loss of energy from the system to its surroun- dings, see [12, 2, 3].

We model the problem of thermoelastic damping in micro-resonators by coupling the plate equation to a modified heat equation with one relaxation parameter proposed by Lord and Shulman [7]. We consider an isotropic and homogenous plate of a uniform

thickness hinR³. The through-thickness displacement at time t >0 is given by

u(x, y, t) :=h⁻¹

h/2

Z

−h/2

u₃(x, y, z, t)dz,

for detailed information see [9, 12, 5]. The displacement satisfies the equation D∆²u+dα(1 +ν)∆θ+ρhu_,tt= 0,

where D, d, α, ν, ρand h are constants. Here,θ is the first thermal moment defined by

θ(x, y, t) =I⁻¹

h/2

Z

−h/2

zη(x, y, z, t)dz,

where η(x, y, z, t) is the temperature.θ is governed by the equation c(θ,t+τ θ,tt)−K∆θ+h

IKθ− αT E

1−2ν∆(u,t+τ u,tt) = 0,

where c is the heat capacity. K, T, E and I are some physical constants. One can find a good review of the relevance of the thermoelastic damping and the derivation for the one dimensional case in [12].

We consider the system in the dimensionless form. The system of equations reads as

a∆²u+ ∆θ+u_tt = F, (1)

∆θ−mθ+d∆ˆu_t = cθˆ_t+G, (2)

where ˆf =f +τ f_t.

We assume first that a, d, c and τ are positive constants. The constant m may be non-negative. F and G correspond to the external force and heat supply. First, we consider a bounded domain B ⊂Rⁿ whose boundary satisfies the assumptions of the divergence theorem. In this paper solutions (u, θ) = (u(x, t), θ(x, t)) with x∈B,t≥0 are considered. The initial conditions are given by

u(x,0) =u₀(x), u_t(x,0) =u₁(x), θ(x,0) =θ₀(x), θ_t(x,0) =θ₁(x) (3) and the boundary conditions read

u(x, t) = ∆u(x, t) =θ(x, t) = 0, x∈∂B, t∈[0,∞) (4) or

u(x, t) =∇u(x, t)·n(x) =θ(x, t) = 0, x∈∂B, t∈[0,∞). (5)

In this context n(x) is the outer normal vector to ∂B at a certainx∈∂B.

This paper consists of two parts. In the first part, we consider the system in a boun- ded domain. Non-exponential stability and exponential stability for a damped system are shown. The resonator equations in the whole space have not been studied in the literature before. In the second part, we consider the system in the whole space. We determine the decay rates of the system.

2 Exponential Stability

The system (1), (2) in a bounded domain with the initial conditions (3) and boundary conditions(4) or (5) was partly considered in [9]. Unfortunately, [9] includes a mistake which was corrected in [10].

2.1 Non-Exponential Stability

The wellposedness was shown in [9]. We consider the thermoelastic system

a∆²u+utt+ ∆θ = F, (6)

∆θ−mθ+d∆ˆu_t = cθˆ_t+G (7)

with the initial conditions (3) and boundary conditions (5). From equation (6) we obtain a∆²uˆ+ ˆutt+ ∆ˆθ= ˆF .

Obviously, we get from (ˆu, θ) the solution (u, θ) of the original system. We define V := (ˆu,uˆ_t, θ, θ_t)⁰.

With the operator A and ˜F given by

A:=











0 1 0 0

−a∆² 0 −∆ −τ∆

0 0 0 1

0 _cτ^d∆ _cτ¹ (∆−m) −¹_τ











, F˜ := (0,F ,ˆ 0,−1 cτG)⁰,

we obtain

V_t=AV + ˜F and V₀(·) := (ˆu,uˆ_t, θ, θ_t)⁰(·,0). (8) We introduce a Hilbert space

H =H₀²(B)×L²(B)×H₀¹(B)×L²(B)

with an inner product

hV, Wi_H := d^DV², W^2E+ad^D∆V¹,∆W^1E+τ^D∇V³,∇W^3E+τ m^DV³, W^3E +c^DV³+τ V⁴, W³+τ W^4E,

where h·,·idenotes the usual inner product in L²(B).

This system is an example for a thermoelastic system which changes its exponential stability to non-exponential stability by changing from Fourier’s law to Cattaneo’s law.

Fourier’s law is given by the equation

q+∇θ= 0,

where q is the heat flux. Replacing Fourier’s law by the equation τ qt+q+∇θ= 0

gives Cattaneo’s law, where τ >0 is the relaxation parameter. It is shown in [10] that the associated semigroupⁿe^tAo

t≥0 forτ >0 is not exponentially stable. This effect can also be observed in some Timoshenko systems, see [11].

2.2 Exponential Stability of a damped System

Next, we want to introduce an additional term, called damping, to assure for the ex- ponential stability. The damped system has the form

a∆²u+ ∆θ+utt+γut = 0, (9)

∆θ−mθ+d∆ˆu_t = cθˆ_t, (10) where γ >0 is a damping factor. The natural energy is given by

E(t) = Z

B

(dˆu²_t +ad|∆ˆu|²+cθˆ²+τ(|∇θ|²+mθ²))dB.

One can show that d

dtE(t) =−2 Z

B

(|∇θ|²+mθ²)dB−2γd Z

B

uˆ²_tdB. (11) Our aim is to determine a suitable Lyapunov functional which is equivalent to the energy. First of all we define

η(x, t) :=

t

Z

0

θ(x, s)ds. (12)

Letting Qbe the solution to

∆Q−mQ= [cθ0+cτ θ1−d∆u0−dτ∆u1] (13) with homogenous boundary conditions Q = 0 on ∂B, we observe that β := η+Q satisfies the homogenized equation

∆β−mβ=cθˆ−d∆ˆu. (14)

It is easy to see thatβ_t=η_t+Q_t=η_t=θholds. Multiplying equation (14) by ˆθyields cθˆ² = ∆βθ+τ∆βθt−mβθ−mτ βθt+d∆ˆuθ.ˆ

Integrating over B, we obtain Z

B

cθˆ²dB = −d dt



 Z

B

1

2(|∇β|²+mβ²) +τ∇β∇β_t+mτ ββt

dB





+ τ Z

B

(|∇θ|²+mθ²)dB+d Z

B

∆ˆuθdB.ˆ (15)

The later integral has to be estimated. We apply the operator "ˆ" to equation (9) and get

a∆²uˆ+ ˆu_tt+ ∆ˆθ+γuˆ_t= 0. (16) We multiply equation (16) with dˆu and find

d Z

B

∆ˆuθdBˆ =− Z

B

ad|∆ˆu|²dB+ Z

B

uˆ²_tdB− d dt

Z

B

dˆu_tudBˆ − d dt

γd 2

Z

B

uˆ²dB. (17)

Altogether we obtain d

dt



 Z

B

1

2(|∇β|²+mβ²) +τ∇β∇β_t+mτ ββ_t+dˆuuˆ_t+dγ 2 uˆ²

dB





=− Z

B

cθˆ²dB− Z

B

ad|∆ˆu|²dB+ Z

B

dˆu²_tdB+τ Z

B

(|∇θ|²+mθ²)dB.

Now we define the Lyapunov functional F(t) :=N E(t) +

Z

B

1

2(|∇β|²+mβ²) +τ∇β∇β_t+mτ ββ_t+dˆuuˆ_t+γd 2 uˆ²

dB,

for a constant N >0 being arbitrary for a moment. It yields d

dtF(t) = −2N Z

B

(|∇θ|²+mθ²)dB−2N γ Z

B

dˆu²_tdB− Z

B

cθˆ²dB

− Z

B

ad|∆ˆu|²dB+ Z

B

dˆu²_tdB+τ Z

B

(|∇θ|²+mθ²)dB

=

−2N+τ τ

τ

Z

B

(|∇θ|²+mθ²)dB+ (−2N γ+ 1) Z

B

dˆu²_tdB

− Z

B

cθˆ²dB− Z

B

ad|∆ˆu|²dB.

We chooseN to satisfy

−2N

τ + 1<0 and −2N γ+ 1<0. (18) Thus, we obtain

d

dtF(t)≤ −min 2N

τ −1,2N γ−1,1

| {z }

=:C

Z

B

(dˆu²_t+ad|∆ˆu|²+cθˆ²+τ(|∇θ|²+mθ²))dB

| {z }

=E(t)

meaning

d

dtF(t)≤ −CE(t), C >0.

Using the homogenized equation (14), one can show the equivalence of the functional and the energy, i.e.,

∃c₁, c₂ >0 : c₁E(t)≤F(t)≤c₂E(t).

In particular, we can show that there is a constant C such that

|F(t)−N E(t)|6CE(t).

For example, we multiply equation (14) withβ. Using Young’s inequality 2ab6a²+¹b² for >0, we get

Z

B

(|∇β|²+mβ²)dB = Z

B

d∆ˆuβdB− Z

B

cθβdBˆ

6

2 Z

B

d²|∆ˆu|²dB+ 1 2

Z

B

β²dB+ 2

Z

B

c²θˆ²dB+ 1 2

Z

B

β²dB

:=_m¹

= 1

2m Z

B

d²|∆ˆu|²dB+ 1 2m

Z

B

c²θˆ²dB+m Z

B

β²dB.

This imply Z

B

|∇β|²dB6 1 2m

Z

B

d²|∆ˆu|²dB+ 1 2m

Z

B

c²θˆ²dB 6CE(t).

Poincaré’s inequality yields Z

B

β²dB 6C Z

B

|∇β|²dB6CE(t).

The remaining terms can be treated similarly. Let us remark thatCis a generic positive constant which does not depend on t. Altogether we obtain

d

dtF(t)≤ −C c₂F(t).

Gronwall’s lemma implies

F(t)≤e⁻

C c2t

F(0).

Therefore,

E(t)≤ 1 c1

F(t)≤ 1 c1

e⁻

C c2t

F(0)≤ c2

c1

E(0)e⁻

C c4t

. This yields the exponential stability of the damped system.

Theorem 1. For the energy of the damped system (9), (10), there exist constants c˜₁ >0 and c˜₂ >0 independent from the initial data such that

E(t)≤˜c₁E(0)e^−˜c2t (19)

holds for all t≥0.

Remark 2. One can similarly show that the following damped system is also exponen- tially stable:

a∆²u+ ∆θ+u_tt−γ∆u_t = 0, (20)

∆θ−mθ+d∆ˆu_t = cθˆ_t. (21)

3 Asymptotics and Decay Rates

The resonator equations in the whole space have not been studied in the literature before. Here, we want to study the asymptotic behaviour of the solutions to the Cauchy problem

a∆²u+ ∆θ+u_tt = 0, (22)

∆θ−mθ+d∆ˆu_t = cθˆ_t, (23)

wheret∈R⁺andx∈Rⁿ. Our aim is to determine the decay rates of this system. Now we have to asssume m6= 0. The initial conditions are

u(x,0) =u0(x), ut(x,0) =u1(x), θ(x,0) =θ0(x), θt(x,0) =θ1(x)

forx∈Rⁿ. We assume the existence of smooth solutions which can be shown by using the Fourier transform. Application of the Fourier transform (F u(x, t))(ξ) =: v(ξ, t), (F θ(x, t))(ξ) =:w(ξ, t) implies

aρ²v(ξ, t)−ρw(ξ, t) +v_tt(ξ, t) = 0, (24) ρw(ξ, t)−mw(ξ, t)−dρˆvt(ξ, t) = cwˆt(ξ, t), (25) where ρ := |ξ|². It is easy to see that both v and w satisfy the following fourth-order ordinary differential equation

v_tttt+ 1

τv_ttt+ 1 cτ

(ac+d)τ ρ²+m+ρv_tt+ 1

cτρ²(ac+d)v_t+ a

cτρ²(ρ+m)v= 0.

(26) The characteristic polynomial of equation (26) is given by

P_ρ(λ) =λ⁴+ 1

τλ³+ 1

cτ(ρ+m+τ(ac+d)ρ²_j)λ²+ 1

cτ(ac+d)ρ²λ+ a

cτ(ρ³+mρ²).

(27) We represent P_ρin the form

P_ρ(λ) =λ⁴+a₃λ³+a₂λ²+a₁λ+a₀.

First, we study the behaviour of the roots for ρ→0. By a straightforward analysis, we obtain the following asymptotic properties of the roots which give information about the decay rates.

Proposition 3. The roots of the characteristic polynomial have the following properties for ξ →0 and ρ:=|ξ|²:

λ1 = −_2m^d ρ²+O(ρ³) +i√

aρ+iO(ρ²), λ₂ = −_2m^d ρ²+O(ρ³)−i√

aρ+iO(ρ²), λ3 = −_2τ¹ +¹_2τ¹2 −^4m_cτ ^1/2− 1

cτ_τ¹2 − ^4m_cτ ^1/2

ρ+O(ρ²),

λ₄ = −_2τ¹ −¹_2τ¹2 −^4m_cτ ^1/2+ 1

cτ_τ¹2 − ^4m_cτ ^1/2

ρ+O(ρ²).

Proof: The roots are explicitly given by

λ1 = −¹₄a3+¹₂R+¹₂D, λ₂ = −¹₄a₃+¹₂R−¹₂D, λ₃ = −¹₄a₃−¹₂R+¹₂E, λ₄ = −¹₄a₃−¹₂R−¹₂E,

where

R = ¹₄a²₃−a2+y1

¹₂

forR6= 0, D = ³₄a²₃−R²−2a₂+

1

4(4a3a2−8a1−a³₃) R

!¹₂ ,

E = ³₄a²₃−R²−2a₂−

1

4(4a3a2−8a1−a³₃) R

!¹₂ ,

where y1 is the real root of the cubic resolvent.

Definition 4. The cubic resolvent of a polynomial of fourth order is given by y³−a₂y²+ (a₁a₃−4a₀)y+ (4a₂a₀−a²₁−a²₃a₀) = 0.

Let ˜a₂,˜a₁,˜a₀ be the coefficients of the cubic resolvent, i.e., y³+ ˜a2y²+ ˜a1y+ ˜a0.

Using Cardano’s formula, we explicitly get the roots of the cubic resolvent y₁=−1

3˜a₂+S+T, where

S = ³

q

R˜+^pD,˜ T = ³ q

R˜−^pD,˜ D˜ =Q³+ ˜R², Q = ^3˜a1₉^−˜a22, R˜ = ^{9˜a2˜a1−27˜}₅₄^a0−2˜a32.

Using Taylor’s expansion ^p1 +ρ+O(ρ²) = 1 + ¹₂ρ+O(ρ²) for ρ→ 0, one can easily prove the claim of the proposition. For example, we obtain

R = 1

2τ − d

mρ²+O(ρ³), D = 2i√

aρ+iO(ρ²).

E =

1 τ² −4m

cτ 1/2

− 2

cτ_τ¹2 −^4m_cτ ^1/2

ρ+O(ρ²)

forρ→0. In particular, we see that we have to assumem6= 0.

The following corollary can be concluded.

Corollary 5. There are constants R > 0 und c1, c2, c3, c4 > 0 depending on R such that

c1ρ² ≤ −Reλ_1,2 ≤c2ρ², c3 ≤ −Reλ_3,4≤c4

holds for ρ≤R.

One can numerically verify these properties. One can also determine the asymptotic behaviour of the roots for ρ→ ∞. The following proposition describes the asymptotic behaviour of the roots for ρ→ ∞.

Proposition 6. The roots of the characteristic polynomial have the following properties for ξ → ∞ and ρ:=|ξ|². There are positive constantsc1, c2, c3, c4>0 such that

λ1,2 = −c₁1 ρ +O

1 ρ²

±c2ρi+iO(ρ²), λ_3,4 = −c₃+O

1 ρ

±c₄√

ρi+iO(ρ).

Proof: As before, we can explicitly calculate the roots of the polynomial. We remark that we can consider ¹_ρ for large ρ, as in the previous proposition. We can use the expansion formula

s 1 + 1

ρ +O( 1

ρ²) = 1 + 1

2ρ +O( 1 ρ²) forρ→ ∞.

As before, we can get the following corollary.

Corollary 7. There are positive constants R >0 and c₁, c₂, c₃, c₄ >0 depending onR such that

c1

1

ρ ≤ −Reλ_1,2 ≤c2

1 ρ, c3 ≤ −Reλ_3,4 ≤c4

holds for ρ≥R.

3.1 Decay Rates for m 6= 0

In this sectionCstands for a generic positive constant. We want to determine the decay rates for the homogenous system (1), (2) for the case m 6= 0. A fundamental system for equation (26) is given by

ne^λ1t, e^λ2t, e^λ3t, e^λ4to. Every solution of equation (26) has the form

v(t, ξ) =a1(ξ)e^λ1t+a2(ξ)e^λ2t+a3(ξ)e^λ3t+a4(ξ)e^λ4t. We can explicitly determine the coefficients aj(ξ) as

a_j(ξ) =

3

X

k=0

b^k_jv_k for j= 1,2,3,4, where b^k_j are given by

b⁰_j =

−Q

l6=j

λ_l

Q

l6=j

λj−λ_l, b¹_j =

P

l6=j6=i

λiλ_l

Q

l6=j

λj−λ_l, b²_j =

−P

l6=j

λ_l

Q

l6=j

λj−λ_l, b³_j = Q ¹

l6=j

λj−λ_l.

The following lemma describes the asymptotic behaviour of the coeffiecients aj(ξ).

Lemma 8. ∃C >0, ∃R₁ >0, ∀ρ≤R₁:

|a₃(ξ)e^λ3(ξ)t| ≤ C(|v₀(ξ)|+|v₁(ξ)|+|v₂(ξ)|+|v₃(ξ)|)e^Reλ3t,

|a₄(ξ)e^λ4(ξ)t| ≤ C(|v₀(ξ)|+|v₁(ξ)|+|v₂(ξ)|+|v₃(ξ)|)e^Reλ4t. Proof: a3(ξ) =

3

P

k=0

b^k₃vk anda4(ξ) =

3

P

k=0

b^k₄vk yielding

|b⁰₃| =

−λ₁λ₂λ₄

(λ₃−λ₁)(λ₃−λ₂)(λ₃−λ₄)

=

−|λ₁|²λ₄

|λ₃−λ₁|²(λ₃−λ₄)

→0 (ξ →0),

|b¹₃| =

λ1λ2+λ1λ4+λ2λ4

|λ₃−λ₁|²(λ₃−λ₄)

→0 (ξ →0),

|b²₃| =

−λ₁+λ2+λ4

|λ₃−λ1|²(λ3−λ4)

→ ^∃C <∞ (ξ→0),

|b³₃| =

1

|λ₃−λ₁|²(λ₃−λ₄)

→ ^∃C <∞ (ξ→0).

b^k₄ can be treated similiarly. These imply the desired claim.

We can also estimate the first part of the solution for small ρ.

Lemma 9. ∃C >0, ∃R₁ >0, ∀ρ≤R1:

|a₁(ξ)e^λ1(ξ)t+a2(ξ)e^λ2(ξ)t| ≤C

|v₀(ξ)|+C|sin(ρt)|

ρ (|v₁(ξ)|+|v₂(ξ)|+|v₃(ξ)|)

e^Reλ1t. Proof: It yields

a₁(ξ)e^λ1(ξ)t+a₂(ξ)e^{λ2(ξ)t λ1}=^=¯λ2 e^Reλ1t

v₀(b⁰₁e^iImλ1t+b⁰₂e^iImλ2t) +· · · + v₃(b³₁e^iImλ1t+b³₂e^iImλ2t)

b^k₁=b^k₂,λ1=¯λ2

= e^Reλ1t

v0(b⁰₁e^iImλ1t+b⁰₁·e^iImλ1t) +· · · + v3(b³₁e^iImλ1t+b³₂·e^iImλ1t)

= e^Reλ1t

v₀·2Re(b⁰₁e^iImλ1t) +· · · + v₃·2Re(b³₁e^iImλ1t)

. On the other hand,

Re(b^k₁e^iImλ1t) = Re(b^k₁)·Re(e^iImλ1t)−Im(b^k₁)·Im(e^iImλ1t)

= Re(b^k₁)·cos(Imλ1t)−Im(b^k₁)·sin(Imλ1t), fork= 0,1,2,3. Using proposition (3) and Taylor’s expansion, we get

b⁰₁ = −λ₂λ3λ4

(λ1−λ2)(λ1−λ3)(λ1−λ4) = −λ₂λ3λ4

O(ρ²) +iO(ρ)

= O(ρ²) +iO(ρ)

O(ρ²) +iO(ρ) =O(1) +iO(1), b¹₁ = λ₂λ₃+λ₂λ₄+λ₃λ₄

O(ρ²) +iO(ρ) = C+O(ρ) +iO(ρ)

O(ρ²) +iO(ρ) =O(1) +iO(ρ⁻¹), b²₁ = −λ₂−λ3−λ4

O(ρ²) +iO(ρ) = C+O(ρ) +iO(ρ)

O(ρ²) +iO(ρ) =O(1) +iO(ρ⁻¹),

b³₁ = 1

O(ρ²) +iO(ρ) =O(1) +iO(ρ⁻¹) for ρ→0.

These yield the desired estimate. As a corollary we obtain Corollary 10. ∃C >0,∃R₁ >0, ∀ρ≤R₁:

|a_i(ξ)e^λi(ξ)t| ≤C|v₀(ξ)|e^Reλit+C1

ρ(|v₁(ξ)|+|v₂(ξ)|+|v₃(ξ)|)e^Reλit, where i= 1,2.

For large ρ, we have

Lemma 11. ∃R₂ >0,∃C >0, ∀ρ≥R2:

|a_ie^λit| ≤Ce^Reλit(|v₀|+|v₁|+|v₂|+|v₃|), where i= 1,2,3,4.

Proof: Analogously we obtain the behaviour for the coefficients using the properties of the roots for ρ→ ∞.

These results lead to the estimate

|v_t| = |λ₁a1e^λ1t+λ2a2e^λ2t+λ3a3e^λ3t+λ4a4e^λ4t|

≤ |λ₁||a₁e^λ1t|+|λ₂||a₂e^λ2t|+|λ₃||a₃e^λ3t|+|λ₄||a₄e^λ4t|.

Let ˜ube the Fourier transformed of v, yielding

|˜ut| = C|

Z

Rⁿ

e^ixξvt(ξ, t)dξ|

≤ C Z

|ξ|≤R₁

|v_t(ξ, t)|dξ+C Z

R1≤|ξ|≤R₂

|v_t(ξ, t)|dξ+C Z

|ξ|≥R₂

|v_t(ξ, t)|dξ

Lemmata (9), (8) and Corollary (5) imply Z

|ξ|≤R₁

|v_t(ξ, t)|dξ≤C(||v₀||_∞+· · ·+||v₃||_∞)e^−Ct+t^−n/4. We easily get

Z

R1≤|ξ|≤R2

|v_t(ξ, t)|dξ≤Ce^−Ct(||v₀||_∞+· · ·+||v₃||_∞). Using

ρe^−ρ1t≤ ρⁿ⁺¹ tⁿ , one can see

Z

|ξ|>R2

|v_t(ξ, t)|dξ ≤ C(||v₀||_3n+3,∞+||v₁||_3n+3,∞+||v₂||_3n+3,∞+||v₃||_3n+3,∞)×

×e^−Ct+t⁻ⁿ. Alltogether we obtain for all t≥0:

|˜ut|6C(1 +t)^−n/4(||u˜₀||_3n+3,1+||˜u₁||_3n+3,1+||˜u₂||_3n+3,1+||˜u₃||_3n+3,1).

We can also get an estimate for the solution in the L²-norm since the system is dissi- pative. Now we can use interpolation techniques to describe the asymptotic behaviour of the solutions.

Theorem 12. Let 2 6q 6∞, 1/p+ 1/q = 1, m 6= 0, Np >(1−2/q)(3n+ 3). Then

∃c=c(n, q)>0 ∀V₀ ∈W^Np,p(Rⁿ) ∀t≥0:

||V_t(t)||_q6c(1 +t)^{−n4(1−2q)}||V₀||_Np,p,

where V(t) = (ˆu(t),uˆt(t), θ(t), θt(t))and (u, θ) the solutions of the system (22), (23).

It should be mentioned that m 6= 0 has been assumed. We will see that the decay rates differ from the case m = 0. Compared to the classical linear thermoelastic plate equations τ = 0, i.e.,

a∆²u+ ∆θ+u_tt = 0, cθ_t−∆θ−d∆u_t = 0,

we also observe different decay rates. Namely, the decay rates in the system of linear thermoelastic plate equation has the form t^−n/2. We remark that the decay ratet^−n/4 for the case m= 0 is not the decay rate of the classical plate equation

u_tt+ ∆²u= 0

having the rate of t^−n/2 corresponding to that of the heat equation. We compare the decay rates of the systems, i.e., we estimate the vector V(t) in some time derivatives to the decay rate. In this case you can do this for first time derivative.

3.2 Decay Rates for m = 0

Letting m= 0, we observe that the homogenized system

a∆²u+ ∆θ+u_tt = 0, (28)

∆θ−mθ+d∆ˆu_t = cθˆ_t (29)

can be rewritten as

a∆²u+ ∆θ+utt = 0, (30)

cθ_t+∇⁰q−d∆u_t = 0, (31)

τ q_t+q+∇θ = 0, (32)

whereq is the heat flux. The system is given by a coupling of the plate equation to the heat equation after Cattaneo’s law. The last equation represents the Cattaneo’s law.

One can easily get for τ = 0 and m= 0 the classical thermoelastic plate equations.

As mentioned before, the system changes its decay rates for m = 0, because the roots of the characteristic polynomial change their behaviour. We assume a=c=d= 1 and m= 0. By a similar analysis we obtain

Proposition 13. There holds for m= 0, ρ→0:

λ1 = 1

2(c1−1)ρ+O(ρ²) +O(ρ²)^1/2, λ2 = 1

2(c1−1)ρ+O(ρ²)−O(ρ²)^1/2, λ₃ = −c₁ρ+O(ρ²),

λ4 = −1

τ +ρ+O(ρ²), where

c1≈0,56984, c1−1≈ −0,43015.

The behaviour of the roots for ρ→ ∞ does not change, so we have the same result as before forρ→ ∞. Analogously, we can study the asymptotics of the system form= 0.

By a similar analysis of the solution as in the section before, we can obtain an estimate for the second time derivative of the solution. So we have the following theorem.

Theorem 14. Let 2 6q 6∞, 1/p+ 1/q = 1, m = 0, Np >(1−2/q)(3n+ 5). Then

∃c=c(n, q)>0 ∀V₀ ∈W^Np,p(Rⁿ) ∀t≥0:

||V_tt(t)||_q6c(1 +t)^{−n2(1−2q)}||V₀||_Np,p, where V(t) = (ˆu(t),uˆt(t), θ(t), θt(t)).

One can get estimates for the vector V(t) putting some conditions on the space di- mension. These theorems are presented next without proofs which we can be done analogously.

Theorem 15. Let 26q6∞, 1/p+ 1/q = 1, m6= 0, N_p >(1−2/q)(3n+ 1), n>3.

Then ∃c=c(n, q)>0 ∀V₀ ∈W^Np,p(Rⁿ) ∀t>0:

||V(t)||_q6c(1 +t)⁻

n−2 4 (1−²_q)

||V₀||_Np,p.

Theorem 16. Let 26q6∞, 1/p+ 1/q = 1, m= 0, N_p >(1−2/q)(3n+ 1), n>5.

Then ∃c=c(n, q)>0 ∀V₀ ∈W^Np,p(Rⁿ) ∀t>0:

||V(t)||_q6c(1 +t)⁻

n−4

2 (1−²_q)||V₀||_Np,p.

Acknowledgement:The present work is based on my diploma thesis at the University of Konstanz. The author thanks the Analysis Research Group at the Department of Mathematics at the University of Konstanz for the helpful and interesting discussions.

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