Under suitable assumptions, we prove the compactness of the dif- ference between the porous thermoelastic semigroup and its decoupled one

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

COMPACTNESS OF THE DIFFERENCE BETWEEN THE POROUS THERMOELASTIC SEMIGROUP AND ITS

DECOUPLED SEMIGROUP

EL MUSTAPHA AIT BENHASSI, JAMAL EDDINE BENYAICH, HAMMADI BOUSLOUS, LAHCEN MANIAR

Abstract. Under suitable assumptions, we prove the compactness of the dif- ference between the porous thermoelastic semigroup and its decoupled one.

This will be achieved by proving the norm continuity of this difference and the compactness of the difference between the resolvents of their generators.

Applications to porous thermoelastic systems are given.

1. Introduction

An increasing interest to determine the decay behavior of solutions of several porous elastic and thermoelastic problems has been discovered recently. The theory of porous elastic material was established first by Cowin and Nunziato [5, 6, 7].

In a recent paper the authors of [25] proved a slow decay of solution of porous elastic system with boundary Dirichlet conditions in one dimensional case. After, Casas and Quintanilla [8], proved the exponential decay of a porous thermoelastic system. This problem has recently been the focus of interest of Glowinsky and Lada [13, 14, 15]. In this work, we consider the abstract porous thermoelastic model

¨

w1(t) +A1w1(t) +C1w2(t) +C2θ(t) = 0, t≥0, (1.1)

¨

w2(t) +A2w2(t)−C₁^∗w1(t)−C3θ(t) +DD^∗w˙2(t) = 0, t≥0, (1.2) θ(t) +˙ A3θ(t)−C₂^∗w˙1(t) +C₃^∗w˙2(t) = 0, t≥0, (1.3) w1(0) =w⁰₁, w˙1(0) =w¹₁, w2(0) =w₂⁰, w˙2(0) =w₂¹, θ(0) =θ⁰, (1.4) with its decoupled system

¨

w1(t) +A1w1(t) +C1w2(t) +C2A⁻¹₃ C₂^∗w˙1(t)−C2A⁻¹₃ C₃^∗w˙2(t)

= 0, t≥0, (1.5)

¨

w₂(t) +A₂w₂(t)−C₁^∗w₁(t)−C₃A⁻¹₃ C₂^∗w˙₁(t) + (C₃A⁻¹₃ C₃^∗+DD^∗) ˙w₂(t)

= 0, t≥0, (1.6)

θ(t) =˙ −A3θ(t) +C₂^∗w˙1(t)−C₃^∗w˙2(t), t≥0, (1.7)

2010Mathematics Subject Classification. 34G10, 47D06.

Key words and phrases. Porous thermoelastic system; semigroup; compactness;

norm continuity; fractional powers; essential spectrum.

c

2015 Texas State University - San Marcos.

Submitted October 14, 2014. Published June 22, 2015.

1

w1(0) =w⁰₁, w˙1(0) =w¹₁, w2(0) =w₂⁰, w˙2(0) =w₂¹, θ(0) =θ⁰. (1.8) The corresponding porous elastic system is given by the first and second equations in the decoupled system (1.5)-(1.8),

¨

w1(t) +A1w1(t) +C1w2(t) +C2A⁻¹₃ C₂^∗w˙1(t)−C2A⁻¹₃ C₃^∗w˙2(t)

= 0, t≥0, (1.9)

¨

w2(t) +A2w2(t)−C₁^∗w1(t)−C3A⁻¹₃ C₂^∗w˙1(t) + (C3A⁻¹₃ C₃^∗+DD^∗) ˙w2(t)

= 0, t≥0, (1.10)

w1(0) =w₁⁰, w˙1(0) =w¹₁, w2(0) =w⁰₂, w˙2(0) =w¹₂. (1.11) In this article, we first show the existence of solution of problems determined by systems (1.1)-(1.4), (1.5)-(1.8) and (1.9)-(1.11) using the Lumer-Phillips theorem from the theory of semigroups [9, Corollary 3.20]]. Second we address the prob- lem of compactness of difference between the porous-thermoelasticityC0-semigroup (T(t))t≥0generated by the system (1.1)-(1.4) and theC0-semigroup (Td(t))t≥0gen- erated by its decoupled system (1.5)-(1.8). As in [1], we prove the norm continuity of t 7−→ T(t)− Td(t) for t > 0, and we show the compactness of the difference R(λ,A)−R(λ,Ad) for everyλ inρ(A)∩ρ(Ad), where Aand Ad are the genera- tors of (T(t))_t≥0and (Td(t))_t≥0, respectively. These two results together with [20, Theorem 2.3] lead to the compactness of the difference T(t)− T_d(t). This yields that the essential spectrums σ_e(T(t)), and σ_e(T_d(t)) coincide. In the case where the operatorsA⁻¹₃ andA^−1/2₁ C1A⁻¹₂ are compact, following a similar argument as in [11], we prove that σe(S(t)) = σe(Td(t)), where (S(t))_t≥0 is the C0-semigroup generated by the system (1.9)-(1.11).

Consequently one can derive stability results on the first semigroup from the ones of the third semigroup. Finally two applications to a porous thermoelastic system are given. In the first application where A⁻¹_i , i = 1,2 are compact but A⁻¹₃ is not compact, we show that only the two essential spectrums σe(T(t)), and σ_e(Td(t)) coincide. The second application is similar to the one given by Glowinsky and Lada in [15], where the exponential stability of porous thermoelastic system is derived from the corresponding decoupled system. In this application, following a different approach and using the compactness of A⁻¹_i , i= 1,2,3, we obtain the same stability result first for the simpler porous elastic system, then the property is derived for the original porous thermoelastic system.

2. Main results

In what follows, A_i : D(Ai) ⊂ H_i → H_i, i = 1,2,3, be self-adjoint positive operators with bounded inverses, andH_ibe Hilbert spaces equipped with the norm k · k_Hi, i= 1,2,3. The operatorA_i can be extended (or restricted) to each H_i,α, such that it becomes a bounded operator

Ai:Hi,α→H_i,α−1, ∀α∈R, (2.1) where forα≥0,Hi,α=D(A^α_i), with the norm kzki,α=kA^α_izkH_i and for α≤0, Hi,α=H_i,−α^∗ , the dual ofH_i,−α with respect to the pivot spaceHi. The operator D∈ L(H2) andD^∗ its adjoint. The coupled operatorsCi, i=1,2,3, satisfy

(C1) D(C₁) ⊂ H₂ → H₁, with adjoint C₁^∗ such that D(A^1/2₂ ) ,→ D(C₁) and D(A^1/2₁ ),→D(C₁^∗).

(C2) D(C₂) ⊂ H₃ → H₁ with adjoint C₂^∗ such that D(A^1/2₃ ) ,→ D(C₂) and D(A^1/2₁ ),→D(C₂^∗).

(C3) D(C₃)⊂H₃→H₂ with adjointC₃^∗such that

D(A^1/2₃ ),→D(C3) and D(A^1/2₂ ),→D(C₃^∗). (2.2) Set

H:=H_1,1/2×H_2,1/2×H₁×H₂×H₃, in this Hilbert space we introduce the new inner product

D

w1

w2

v1

v2

θ

! ,





we₁ wf2

ev₁ ev₂ eθ



 E

=hw₁,we₁i_H1,1/2+hw₂,wf₂i_H2,1/2+hv₁,ev₁i_H1+hv₂,ev₂i_H2 +hθ,θieH₃+<(hC₁^∗w1,we2iH₂− hw2, C₁^∗we1iH₂).

The associated norm of this inner product coincides with the canonical norm ofH.

We can rewrite (1.1)-(1.4) and (1.5)-(1.8) as the first order evolution equations inH,

dη

dt =Aη, η∈ H, η(0) = (w₁⁰, w⁰₂, w¹₁, w₂¹, θ⁰), and

dη

dt =Adη, η∈ H, η(0) = (w₁⁰, w⁰₂, w¹₁, w₂¹, θ⁰),

respectively, whereAis the unbounded linear operator defined by A:D(A)⊂ H → H, A=





0 0 I 0 0

0 0 0 I 0

−A1−C1 0 0 −C2

C₁^∗ −A2 0 −DD^∗ C₃ 0 0 C^∗₂ −C₃^∗ −A3



, (2.3)

with

D(A) =D(A1)× D(A2)× D(A^1/2₁ )× D(A^1/2₂ )× D(A3), (2.4) and the operatorA_d associated to the decoupled system

Ad:D(Ad) =D(A)⊂ H → H,Ad

=







0 0 I 0 0

0 0 0 I 0

−A1 −C1−C2A⁻¹₃ C₂^∗ C₂A⁻¹₃ C^∗₃ 0 C₁^∗ −A2 C3A⁻¹₃ C₂^∗ −C3A⁻¹₃ C₃^∗−DD^∗ 0

0 0 C^∗₂ −C^∗₃ −A3





. (2.5)

We rewrite the coupled second order system (1.9)-(1.11) on the Hilbert space Hc:=H_1,1/2×H_2,1/2×H1×H2,

as the first order evolution equation deη

dt =Mη,e eη∈ H_c, ηe⁰= (w₁⁰, w⁰₂, w₁¹, w¹₂),

andM:D(M)⊂ Hc→ Hc, is the unbounded linear operator defined by M=

0 0 I 0

0 0 0 I

−A1 −C1−C2A⁻¹₃ C₂^∗ C₂A⁻¹₃ C^∗₃ C₁^∗ −A2 C3A⁻¹₃ C₂^∗ −C3A⁻¹₃ C₃^∗−DD^∗

!

, (2.6)

with

D(M) =D(A1)× D(A2)× D(A^1/2₁ )× D(A^1/2₂ ). (2.7) Now we formulate the main results of this paper.

Theorem 2.1. The operatorsA,AdandMgenerate strongly continuous contrac- tion semigroups(T(t))t≥0,(Td(t))t≥0 onHand(S(t))t≥0 onHc.

Theorem 2.2. Assume that

A^−1/2₁ C2A⁻¹₃ , A^−1/2₁ C1A⁻¹₂ , A^−1/2₂ C3A⁻¹₃ , (2.8) are compact operators from H3 toH1, from H2 toH1 and from H3 toH2 respec- tively. Then T(t)− Td(t)is compact for every t≥0.

As a consequence of Theorem 2.2, we have the following particular results.

Corollary 2.3. Assume that the operators A⁻¹_i , i = 1,2, are compact. Then T(t)− Td(t)is compact for everyt≥0.

Corollary 2.4. Assume that the operators A⁻¹₃ and A^−1/2₁ C1A⁻¹₂ are compact.

Thenσe(T(t)) =σe(S(t))fort≥0.

3. Well-posedness results

In this section we use Lumer-Phillips theorem (see [9, Corollary 3.20]) for the proof of Theorem 2.1.

3.1. Porous thermoelastic system. To show that the operator (A,D(A)) de- fined by (2.3)-(2.4) generates a contraction semigroup on the Hilbert H, we need the following technical lemma.

Lemma 3.1. The operator Ais invertible inHandA⁻¹ is bounded onH.

Proof. Given a vector





f1

f₂ f3

f₄ f5



∈ H, we need

w1

w2

v₁ v₂ w₃

!

∈ D(A), such that

A

w₁ w₂ v₁ v₂ w₃

!

=





f₁ f2

f₃ f4

f₅



. We have

A

w₁ w₂ v₁ v₂ w₃

!

=





f₁ f2

f₃ f4

f₅



⇔



















v1=f1, v₂=f₂,

A1w1+C1w2+C2w3=−f3,

−C₁^∗w₁+A₂w₂+DD^∗v₂−C₃w₃=−f4,

−C₂^∗v1+C₃^∗v2+A3w3=−f5.

Hence

A

w₁ w₂ v₁ v₂ w₃

!

=





f₁ f2

f₃ f4

f₅



⇔



















v1=f1, v₂=f₂,

A1w1+C1w2+C2w3=−f3,

−C₁^∗w₁+A₂w₂−C₃w₃=−f₄−DD^∗f₂, w3=−A⁻¹₃ (f5−C₂^∗f1+C₃^∗f2),

⇔



















v₁=f₁, v2=f2,

A₁w₁+C₁w₂=C₂A⁻¹₃ (f₅−C₂^∗f₁+C₃^∗f₂)−f₃=K₁,

−C₁^∗w1+A2w2=−C3A⁻¹₃ (f5−C₂^∗f1+C₃^∗f2)−f4−DD^∗f2=K2, w₃=−A⁻¹₃ (f₅−C₂^∗f₁+C₃^∗f₂),

⇔



















v1=f1, v₂=f₂,

w1=−A⁻¹₁ C1w2+A⁻¹₁ K1,

(C₁^∗A⁻¹₁ C₁+A₂)w₂=K₂+C₁^∗A⁻¹₁ K₁, w3=−A⁻¹₃ (f5−C₂^∗f1+C₃^∗f2).

We have

v1=f1∈H_1,1/2, v2=f2∈H_2,1/2, w3=−A⁻¹₃ (f5−C₂^∗f1+C₃^∗f2)∈ D(A3).

Suppose that we have foundw₂ with the appropriate regularity. Then, w1=−A⁻¹₁ C1w2+A⁻¹₁ K1∈ D(A1).

We now solve the equation

(C₁^∗A⁻¹₁ C1+A2)w2=K2+C₁^∗A⁻¹₁ K1. (3.1) To findw₂we introduce a bilinear form Λ on D(A^1/2₂ ), defined by

Λ(η, ζ) =hA^−1/2₁ C1η, A^−1/2₁ C1ζi+hA₂¹²η, A

1 2

2ζi.

Since Λ is a bilinear continuous and coercive form on D(A^1/2₂ ), the Lax-Milgram Lemma leads to the existence and uniqueness of w₂ ∈ D(A^1/2₂ ) solution to the equation (3.1).

Moreover K2 +C₁^∗A⁻¹₁ K1−C₁^∗A⁻¹₁ C1w2 ∈ H2 and [(A2)−1]⁻¹H2 = D(A2), (where (A2)−1 is an extension of A2), then w2 ∈ D(A2), (see [3, Proposision 5]).

SetB1= (C₁^∗A⁻¹₁ C1+A2)⁻¹, then we have

A

w₁ w₂ v₁ v₂ w₃

!

=





f₁ f2

f₃ f4

f₅



⇔



















v1=f1, v₂=f₂,

w1=−A⁻¹₁ C1w2+A⁻¹₁ K1, w₂=B₁K₂+B₁C₁^∗A⁻¹₁ K₁, w3=−A⁻¹₃ (f5−C₂^∗f1+C₃^∗f2),

⇔















































v1=f1, v₂=f₂,

w1= (−A⁻¹₁ C1B1C3A⁻¹₃ C₂^∗+A⁻¹₁ C1B1C₁^∗A⁻¹₁ C2A⁻¹₃ C₂^∗

−A⁻¹₁ C₂A⁻¹₃ C₂^∗)f₁+ (A⁻¹₁ C₁B₁C₃A⁻¹₃ C₃^∗+A⁻¹₁ C₁B₁DD^∗

−A⁻¹₁ C1B1C₁^∗A⁻¹₁ C2A⁻¹₃ C₃^∗+A⁻¹₁ C2A⁻¹₃ C₃^∗)f2

+(A⁻¹₁ C₁B₁C₁^∗A⁻¹₁ −A⁻¹₁ )f₃+A⁻¹₁ C₁B₁f₄

+(A⁻¹₁ C1B1C3A⁻¹₃ −A⁻¹₁ C1B1C₁^∗A⁻¹₁ C2A⁻¹₃ +A⁻¹₁ C2A⁻¹₃ )f5, w₂= (B₁C₃A⁻¹₃ C₂^∗−B₁C₁^∗A⁻¹₁ C₂A⁻¹₃ C₂^∗)f₁

+(−B1C3A⁻¹₃ C₃^∗−B1DD^∗+B1C₁^∗A⁻¹₁ C2A⁻¹₃ C₃^∗)f2

−B1C₁^∗A⁻¹₁ f₃−B₁f₄+ (−B1C₃A⁻¹₃ +B₁C₁^∗A⁻¹₁ C₂A⁻¹₃ )f₅, w3=−A⁻¹₃ (f5−C₂^∗f1+C₃^∗f2).

Thus,

A⁻¹=







a₁₁ a₁₂ a₁₃ A⁻¹₁ C₁B₁ a₁₅ a21 a22 −B1C₁^∗A⁻¹₁ −B1 a25

I 0 0 0 0

0 I 0 0 0

A⁻¹₃ C₂^∗−A⁻¹₃ C₃^∗ 0 0 −A⁻¹₃





, (3.2)

where

a11=−A⁻¹₁ C1B1C3A⁻¹₃ C₂^∗+A⁻¹₁ C1B1C₁^∗A⁻¹₁ C2A⁻¹₃ C₂^∗−A⁻¹₁ C2A⁻¹₃ C₂^∗, a12=A⁻¹₁ C1B1C3A⁻¹₃ C₃^∗+A⁻¹₁ C1B1DD^∗−A⁻¹₁ C1B1C₁^∗A⁻¹₁ C2A⁻¹₃ C₃^∗

+A⁻¹₁ C2A⁻¹₃ C₃^∗,

a₁₃=A⁻¹₁ C₁B₁C₁^∗A⁻¹₁ −A⁻¹₁ ,

a₁₅=A⁻¹₁ C₁B₁C₃A⁻¹₃ −A⁻¹₁ C₁B₁C₁^∗A⁻¹₁ C₂A⁻¹₃ +A⁻¹₁ C₂A⁻¹₃ , a21=B1C3A⁻¹₃ C₂^∗−B1C₁^∗A⁻¹₁ C2A⁻¹₃ C₂^∗,

a22=−B1C3A⁻¹₃ C₃^∗−B1DD^∗+B1C₁^∗A⁻¹₁ C2A⁻¹₃ C₃^∗, a₂₅=−B₁C₃A⁻¹₃ +B₁C₁^∗A⁻¹₁ C₂A⁻¹₃ .

The boundedness of the operatorA⁻¹follows by the assumptions (2.2).

Now, to prove that the operatorAgenerates a strongly continuous contraction semigroup (T(t))_t≥0 on H, we have only to show that (A,D(A)) is a dissipative operator onHandλI− Ais surjective for some λ >0.

For every

w1

w2

v1

v2

w3

!

∈ D(A), by the Cauchy-Schwartz inequality, we have

<

A

w₁ w₂ v₁ v₂ w₃

! ,

w₁ w₂ v₁ v₂ w₃

!

=<





v1

v2

−A1w1−C1w2−C2w3

C₁^∗w₁−A2w₂−DD^∗v₂+C₃w₃ C₂^∗v1−C₃^∗v2−A3w3



,

w₁ w₂ v₁ v₂ w₃

!

=<

hv1, w₁iH_1,1/2+hv2, w₂iH_2,1/2− hA1w₁, v₁iH1− hC1w₂, v₁iH1

− hC2w₃, v₁iH1+hC₁^∗w₁, v₂iH2− hA2w₂, v₂iH2− hDD^∗v₂, v₂iH2

+hC3w3, v2iH₂+hC₂^∗v1, w3iH₃− hC₃^∗v2, w3iH₃− hA3w3, w3iH₃

+hC₁^∗v1, w2iH₂−< v2, C₁^∗w1iH₂

=−kD^∗v₂k²_H

2− kA^1/2₃ w₃k²_H

3 ≤0.

Finally, A is dissipative. By a standard argument, one shows that (λI − A) is surjective forλ∈(0,_kA¹−1k). Thus, [9, Corollary 3.20] leads to the claim.

3.2. Decoupled system. We show that the operator (Ad,D(Ad)), associated with the decoupled system (1.5)-(1.8), generates a contraction semigroup on the Hilbert spaceH. For this, we first show the following lemma.

Lemma 3.2. The operator Ad is boundedly invertible inH.

Proof. Following the argument of the proof of Lemma 3.1, we show that the oper- atorAdis invertible and

A⁻¹_d =













b11 b12 b13 A⁻¹₁ C1B1 0 b21 b22 −B1C₁^∗A⁻¹₁ −B1 0

I 0 0 0 0

0 I 0 0 0

A⁻¹₃ C₂^∗ −A⁻¹₃ C₃^∗ 0 0 −A⁻¹₃













, (3.3)

where

b11=−A⁻¹₁ C1B1C3A⁻¹₃ C₂^∗+A⁻¹₁ C1B1C₁^∗A⁻¹₁ C2A⁻¹₃ C₂^∗−A⁻¹₁ C2A⁻¹₃ C₂^∗, b₁₂=A⁻¹₁ C₁B₁C₃A⁻¹₃ C₃^∗+A⁻¹₁ C₁B₁DD^∗−A⁻¹₁ C₁B₁C₁^∗A⁻¹₁ C₂A⁻¹₃ C₃^∗

+A⁻¹₁ C2A⁻¹₃ C₃^∗,

b13=A⁻¹₁ C1B1C₁^∗A⁻¹₁ −A⁻¹₁ , b21=B1C3A⁻¹₃ C₂^∗−B1C₁^∗A⁻¹₁ C2A⁻¹₃ C₂^∗, b₂₂=−B₁C₃A⁻¹₃ C₃^∗−B₁DD^∗+B₁C₁^∗A⁻¹₁ C₂A⁻¹₃ C₃^∗.

Now we show the dissipativity of the operator (Ad,D(Ad)) onH. Take

w1

w2

v1

v2

w3

!

∈ D(A_d), by the Cauchy-Schwarz inequality, we have

<

A_d

w₁ w₂ v₁ v₂ w₃

! ,

w₁ w₂ v₁ v₂ w₃

!

=<

hv1, w1iH_1,1/2+hv2, w2iH_2,1/2− hA1w1, v1iH₁

− hC1w2, v1iH₁− hC2A⁻¹₃ C₂^∗v1, v1iH₁+hC2A⁻¹₃ C₃^∗v2, v1iH₁

+hC₁^∗w1, v2iH₂− hA2w2, v2iH₂+hC3A⁻¹₃ C₂^∗v1, v2iH₂

− h(C₃A⁻¹₃ C₃^∗+DD^∗)v₂, v₂i_H2+hC₂^∗v₁, w₃i_H3−< C₃^∗v₂, w₃i_H3

− hA3w3, w3iH₃+hC₁^∗v1, w2iH₂−< C₁^∗w1, v2iH₂

=<

− kD^∗v₂k²_H2− kA^1/2₃ C₂^∗v₁k²_H3+ 2hA^−1/2₃ C₃^∗v₂, A^−1/2₃ C₂^∗v₁i

− kA^−1/2₃ C₃^∗v₂k²

≤ −kD^∗v2k²_H2≤0.

The proof ofλI− Ais surjective for some λ >0, follows as in Theorem 2.1.

3.3. Porous elastic system. As above, we can compute the operator

M⁻¹=









b₁₁ b₁₂ b₁₃ A⁻¹₁ C₁B₁ b21 b22 −B1C₁^∗A⁻¹₁ −B1

I 0 0 0

0 I 0 0









, (3.4)

whereB1= (C₁^∗A⁻¹₁ C1+A2)⁻¹, and

b11=−A⁻¹₁ C1B1C3A⁻¹₃ C₂^∗+A⁻¹₁ C1B1C₁^∗A⁻¹₁ C2A⁻¹₃ C₂^∗−A⁻¹₁ C2A⁻¹₃ C₂^∗, b₁₂=A⁻¹₁ C₁B₁C₃A⁻¹₃ C₃^∗+A⁻¹₁ C₁B₁DD^∗−A⁻¹₁ C₁B₁C₁^∗A⁻¹₁ C₂A⁻¹₃ C₃^∗

+A⁻¹₁ C₂A⁻¹₃ C₃^∗,

b13=A⁻¹₁ C1B1C₁^∗A⁻¹₁ −A⁻¹₁ , b21=B1C3A⁻¹₃ C₂^∗−B1C₁^∗A⁻¹₁ C2A⁻¹₃ C₂^∗, b₂₂=−B₁C₃A⁻¹₃ C₃^∗−B₁DD^∗+B₁C₁^∗A⁻¹₁ C₂A⁻¹₃ C₃^∗,

and show that the operator Mgenerates a strongly continuous contraction semi- group (S(t))t≥0 onHc.

4. Compactness result

In this section we prove the compactness of the differenceT(t)−Td(t), we use [20, Theorem 2.3], where it is sufficient to prove the norm continuity of the difference between the two semigroups, and the compactness of the difference between the resolvents of their generators. To show the first assertion, we need the following technical lemma, see [21, Theorem 1.4.3] .

Lemma 4.1. The map t7→A^α₃e^−A3t is norm continuous on(0,∞)for allα≥0.

Now we can show the following norm continuity result.

Theorem 4.2. The map t7→ T(t)− Td(t)is norm continuous on(0,∞).

Proof. Lett >0 andx0=









w⁰₁ w⁰₂ w¹₁ w¹₂ w⁰₃









∈ D(A) such thatkx0k ≤1. Let us write

T(t)x0− Td(t)x0=







w1(t)−w1(t) w₂(t)−w2(t) v₁(t)−v₁(t) v2(t)−v2(t) w₃(t)−w3(t)





= Z t

0

T(t−s)





0 0 f(s) g(s) 0



ds, where

f(s) =C2A⁻¹₃ C₂^∗v1(s)−C2A⁻¹₃ C₃^∗v2(s)−C2w3(s), g(s) =−C₃A⁻¹₃ C₂^∗v₁(s) +C₃A⁻¹₃ C₃^∗v₂(s) +C₃w₃(s).

Let 0< h <1, we begin by checking thatkf(s+h)−f(s)k →0 ash→0.

We have w₃(t) = e^−A3tw⁰₃+Rt

0e^{−A3(t−σ)}C₂^∗v₁(σ)dσ−Rt

0e^{−A3(s−σ)}C₃^∗v₂(σ)dσ.

Then

f(s) =C2A⁻¹₃ C₂^∗v1(s)−C2A⁻¹₃ C₃^∗v2(s)−C2e^−A3sw₃⁰

−C2A^−1/2₃ Z s

0

A^1/2₃ e^{−A3(s−σ)}C₂^∗v1(σ)dσ

+C2A^−1/2₃ Z s

0

A^1/2₃ e^{−A3(s−σ)}C₃^∗v2(σ)dσ

= (C2A^−1/2₃ )(C2A^−1/2₃ )^∗v1(s)−(C2A^−1/2₃ )(C3A^−1/2₃ )^∗v2(s)

−(C₂A^−1/2₃ )A^1/2₃ e^−A3sw₃⁰−(C₂A^−1/2₃ ) Z s

0

A₃e^{−A3(s−σ)}(C₂A^−1/2₃ )^∗v₁(σ)dσ + (C2A^−1/2₃ )

Z s 0

A3e^{−A3(s−σ)}(C3A^−1/2₃ )^∗v2(σ)dσ.

Since C₂A^−1/2₃ and C₃A^−1/2₃ are bounded operators from H₃ to H₁ and from H₃ toH₂respectively, ands7→e^−A3s,s7→A^1/2₃ e^−A3sare norm continuous on (0,∞), the maps7→(C₂A^−1/2₃ )A^1/2₃ e^−A3(s)is norm continuous on (0,∞), and there exists a positive constantα(s) andβ(s) such thatk(C2A^−1/2₃ )^∗v1(σ)k ≤α(s)kv1(σ)kand k(C3A^−1/2₃ )^∗v2(σ)k ≤β(s)kv2(σ)k, for everyσ∈[0, s). By the inequality

k(w1, w2, v1, v2, w3)k_H≤ kx0k_H, for allt≥0, we deduce

k(C2A^−1/2₃ )^∗v1(σ)k ≤α(s)kx0k, k(C3A^−1/2₃ )^∗v2(σ)k ≤β(s)kx0k, for everyσ∈[0, s).Thus

s7→

Z s 0

A3e^{−A3(s−σ)}(C2A^−1/2₃ )^∗v1(σ)dσ, s7→

Z s 0

A3e^{−A3(s−σ)}(C3A^−1/2₃ )^∗v2(σ)dσ are continuous on (0,∞) uniformly with respect to kx0k ≤1.

Finally kf(s+h)−f(s)k → 0, as h → 0, uniformly in x0. Using the same argument, we havekg(s+h)−g(s)k →0, ash→0, uniformly inx0.

Let us write







w₁(t+h)−w1(t+h) w₂(t+h)−w₂(t+h) v1(t+h)−v1(t+h) v₂(t+h)−v2(t+h) w₃(t+h)−w₃(t+h)





−







w₁(t)−w1(t) w₂(t)−w₂(t) v1(t)−v1(t) v₂(t)−v2(t) w₃(t)−w₃(t)







=

Z t+h 0

T(t+h−s)





0 0 f(s) g(s) 0



ds− Z t

0

T(t−s)





0 0 f(s) g(s) 0



ds

=

Z t+h 0

T(s)





0 0 f(t+h−s) g(t+h−s)

0



ds− Z t

0

T(s)





0 0 f(t−s) g(t−s)

0



ds

=

Z t 0

T(s)





0 0 f(t+h−s)−f(t−s) g(t+h−s)−g(t−s)

0



ds+ Z h

0

T(t+s)





0 0 f(h−s) g(h−s)

0



ds

≤

Z t 0





0 0 f(t+h−s)−f(t−s) g(t+h−s)−g(t−s)

0



ds +

Z h 0





0 0 f(h−s) g(h−s)

0



ds .