Long-time dynamics of an extensible plate equationwith thermal memory

Alisson Rafael Aguiar Barbosa∗

To Fu Ma†

Instituto de Ciencias Matematicas e de Computacao, Universidade de Sao Paulo

13566-590 Sao Carlos, Sao Paulo, Brazil

Abstract

This work is concerned with long-time dynamics of solutions of extensible plateequations with thermal memory. The problem corresponds to a model of ther-moelasticity based on a theory of non-Fourier heat flux. By considering the casewhere rotational inertia is present we show that the thermal dissipation is sufficientto stabilize the system and guarantee the existence of a finite-dimensional globalattractor. In addition the existence of exponential attractors is also considered.

1 Introduction

This paper is concerned with long-time behavior of solutions of a class of nonlinearproblems modeling thermoelastic extensible plates with non-Fourier thermal conductionlaws. The model is written as

utt +∆2u−M(∫Ω|∇u|2dx)∆u−∆utt + f(u) + ν∆θ = h in Ω× R+, (1.1)

θt − ω∆θ − (1− ω)

∫ ∞

0

k(s)∆θ(t− s)ds− ν∆ut = 0 in Ω× R+, (1.2)

where Ω is a bounded domain of R2 with regular boundary Γ. We assume boundaryconditions

u = ∆u = 0 on Γ× R+, (1.3)

θ = 0 on Γ× R, (1.4)

and initial conditions

u(x, 0) = u0(x), ut(x, 0) = u1(x), θ(x, t)|t≤0 = θ0(x,−t), x ∈ Ω, (1.5)

where θ0(x, t) is prescribed for t ≤ 0.

∗PhD fellow at ICMC-USP. Email: alissonrafael@yahoo.com.br.†Corresponding author. Email: matofu@icmc.usp.br.

1

When M is a increasing linear function and ∆utt is not present, the equation (1.1)models transversal vibrations of thin extensible elastic plates, which in its rest configura-tion, occupies a planar region Ω. Modeling aspects of extensible beams and plates wereconsidered by Woinowsky-Krieger [32], Berger [3] and also by Giorgi et al [18]. With-out thermal effects, this class of plate equations were studied by several authors, e.g.[2, 4, 11, 12, 18, 25, 26, 34].

In the case ω = 1, equation (1.2) becomes the classical parabolic heat equation satis-fying the Fourier law of heat conduction. Now if 0 ≤ ω < 1, then equation (1.2) modelshyperbolic type heat flows of finite speed. Indeed, it is usually divided into two subcases.For ω = 0 it corresponds to the heat law of Gurtin and Pipkin [22] and when 0 < ω < 1it corresponds to that of Coleman and Gurtin [8]. We refer the reader to Straughan [30]for a more recent review on heat waves.

Let us make some comments about previous works on the long-time dynamics ofthermoelastic plates of hyperbolic type. Firstly we consider the problem under the absenceof rotational inertia −∆utt. In Grasselli et al [21], it was proved that the linear systemcorresponding to (1.1)-(1.5) is not exponentially stable for a large class of kernels k. Theystudied (1.1)-(1.5) with M = f = h = 0 = ω = 0 by using semigroup theory. This wasalso proved, for more general coupling terms, by Coti Zelati et al [9]. With respect tothe existence of global attractors, still without rotational inertia, Wu [33] considered thesystem (1.1)-(1.5) with M = 0. Then by adding a structural damping −∆ut in (1.1) heproved the existence of a global attractor and also the convergence of global solutionsto equilibrium. In the same direction, Potomkin [29] considered the nonlinear system(1.1)-(1.5) with f = 0. He proved the existence of global and exponential attractors byadding a memory term in the equation (1.1).

On the other hand, Grasselli and Squassina [20] proved, via energy methods, thatthe presence of rotational inertia −∆ut restores the exponential stability of the linearsystem corresponding to (1.1)-(1.5). To the authors best knowledge, the existence ofglobal attractors to thermoelastic plates of hyperbolic type was not considered in thepresence of rotational inertia.

Motivated by the above results we propose to study the existence of global and expo-nential attractors to the problem (1.1)-(1.5) with all its nonlinear terms and with the soledissipation given by the thermal memory. Then our study essentially complements theone of Wu [33] and that of Potomkin [29] to the case with rotational inertia. It also com-plements the results of Grasselli and Squassina [20] to the study of global attractors. Alsoextends some results in Giorgi et al [16, 20]. We emphasize that no additional dampingin the equation (1.1) is need in the present work.

It is well known that, because of the memory term with past history, problems like(1.1)-(1.5) does not correspond to an autonomous system. Then we proceed as in Giorgi,Marzotti and Pata [15] and Giorgi and Pata [17], and define a new variable η = ηt(x, s)by

ηt(s) =

∫ s

0

θ(t− τ)dτ, (t, s) ∈ [0,∞)× R+. (1.6)

2

This corresponds to the summed past history of θ and formally satisfies

ηt + ηs = θ in Ω, (t, s) ∈ R+ × R+,

withηt(0) = 0 in Ω, t ∈ R+,

andη0(s) = η0(s) in Ω, s ∈ R+,

where

η0(s) =

∫ s

0

θ0(τ)dτ, s ∈ R+.

Now let us writeµ(s) = −(1− ω)k′(s),

where 0 ≤ ω < 1 by hypothesis. Then from (1.6) we have∫ ∞

0

k(s)θ(t− s)ds = −∫ ∞

0

k′(s)ηt(s)ds,

and therefore

(1− ω)

∫ ∞

0

k(s)∆θ(t− s)ds =

∫ ∞

0

µ(s)∆ηt(s)ds. (1.7)

From this, problem (1.1)-(1.5) is transformed into the new system

utt +∆2u−M(∥∇u∥22)∆u−∆utt + f(u) + ν∆θ = h(x), (1.8)

θt − ω∆θ −∫ ∞

0

µ(s)∆ηt(s)ds− ν∆ut = 0, (1.9)

ηt + ηs = θ, (1.10)

with boundary conditions

u = ∆u = 0 on Γ× R+, θ = 0 on Γ× R+, ηt(·, 0) = 0 on Γ× R+ × R+, (1.11)

and initial conditions

u(x, 0) = u0(x), ut(x, 0) = u1(x), θ(x, 0) = θ0(x), η0(x, s) = η0(x, s), (1.12)

whereu0(x) = u0(x, 0), u1(x) = ∂tu0(x, 0)|t=0, θ0(x) = θ0(x, 0),

η0(x, s) = u0(x, 0)− u0(x,−s).Our assumptions and results are presented with respect to this new system (1.8)-(1.12)which is now autonomous in an extended phase space containing the history variable η.

The paper is organized in the following way. In Section 2 we present our hypothesesand prove an existence result for the system (1.8)-(1.12), namely, Theorem 2.1. In Section3 we present our main result Theorem 3.1 on the existence of a global attractor with finite-fractal dimension. In Section 4 we consider the existence of exponential attractors. Themain result is Theorem 4.1.

3

2 Assumptions and well-posedness

Let λ1 > 0 be the first eigenvalue of −∆ with boundary condition u = 0 on Γ. Thenwe assume M ∈ C1(R) and there exist constants α ∈ [0, λ1) and cM ≥ 0 such that

M(s)s ≥ −αs− cM and M(s) ≥ −αs− cM , ∀ s ≥ 0, (2.13)

where M(z) =∫ z

0M(s) ds. A typical M satisfying the above conditions is

M(s) = as+ b, a ≥ 0, b > −λ1.

Here we do not require a condition like M(s)s ≥ 12M(s). With respect to the forcing

terms h and f we assume that

h ∈ L2(Ω) and f ∈ C1(R), (2.14)

and there exist r ≥ 1 and Cf > 0 such that

|f ′(s)| ≤ Cf (1 + |s|r−1), ∀ s ∈ R. (2.15)

We also assume there exist constants β ∈ [0, λ21) and cf > 0 such that

f(s)s ≥ −βs2 − cf and f(s) ≥ −β2s2 − cf , ∀ s ∈ R, (2.16)

where f(z) =∫ z

0f(s) ds. A typical assumption that implies the above conditions is

lim inf|s|→∞

f(s)

s> −λ21.

In addition, we needα

λ1+β

λ21< 1. (2.17)

With respect to the memory kernel we assume

µ ∈ C1(R+) ∩ L1(R) ∩ C[0,∞), µ ≥ 0, (2.18)

and there exists k1 > 0 such that

µ′(s) ≤ −k1µ(s), s ≥ 0. (2.19)

In particular from (2.18) we can define the constant

k0 =

∫ ∞

0

µ(s)ds. (2.20)

We will consider the following function spaces.

V0 = L2(Ω), V1 = H10 (Ω), V2 = H2(Ω) ∩H1

0 (Ω),

4

V3 = u ∈ H3(Ω) |u = ∆u = 0 on Γ.With respect to the new variable η we define the weighted space

M = L2µ(R+, V1) =

η : R+ → V1

∣∣∣∣∫ ∞

0

µ(s)∥η(s)∥2V1ds

,

which is a Hilbert space with inner-product defined by

⟨η, ζ⟩M =

∫ ∞

0

µ(s)

∫Ω

∇η(s)∇ζ(s) dxds.

Then our analysis is made on the phase space

H = V2 × V1 × V0 ×M.

Since we apply semigroup theory to prove the well-posedness of (1.8)-(1.12) we considerits equivalent Cauchy problem. To this end we first remark that derivative ηs can bewritten in an operator form. Indeed, it is shown in Grasselli and Pata [19] that operator

Tη = −ηs, η ∈ D(T )

withD(T ) = η ∈ M| ηs ∈ M, η(0) = 0,

is the infinitesimal generator of a translation semigroup. In particular,

⟨Tη, η⟩M =

∫ ∞

0

µ′(s)∥∇η(s)∥22 ds, η ∈ D(T ). (2.21)

and the solutions ofηt = Tη + θ, η(0) = 0,

has an explicit representation formula.

Using notation z = (u, v, θ, η)T the system (1.8)-(1.12) is equivalent to

d

dtz(t) = Lz(t) + F (z(t)), z(0) = z0 ∈ H, (2.22)

where

L =

0 I 0 0

−(I −∆)−1∆2 0 −ν(I −∆)−1∆ 00 ν∆ ω∆ Bµ

0 0 I T

(2.23)

and

Bµη =

∫ ∞

0

µ(s)∆η(s) ds,

with domain

D(L) =

(u, v, θ, η) ∈ H

∣∣∣∣∣∣v ∈ V2, η ∈ D(T ), θ ∈ V1,∆u− νθ ∈ V2,ω∆θ +Bµη ∈ V0.

(2.24)

5

and

F (z) = ( 0,M(∥∇u∥22)(I −∆)−1∆u− (I −∆)−1(f(u)− h)), 0, 0 )T . (2.25)

Our first result is the following.

Theorem 2.1. Let us assume that hypotheses (2.13)-(2.19) hold. Then we have the fol-lowing assertions.

(i) If z0 ∈ H then problem (1.8)-(1.12) has a unique mild solution z ∈ C(0, T ;H) withz(0) = z0, for any T > 0.

(ii) If z1 and z2 are two mild solutions of problem (1.8)-(1.12) then there exists a constantc0 > 0, depending on initial data, such that

∥z1(t)− z2(t)∥H ≤ ec0T∥z1(0)− z2(0)∥H, 0 ≤ t ≤ T. (2.26)

In particular, the mild solutions depend continuously on the initial data in H.

(iii) If z0 ∈ D(L) then the above mild solution is a strong solution.

Remark 2.1. Theorem 2.1 implies that the solution operator

S(t) : H → H, S(t)(u0, u1, θ0, η0) = (u(t), ut(t), θ(t), ηt),

where (u, ut, θ, η) is the unique solution (1.8)-(1.12) with initial data (u0, u1, θ0, η0), satis-fies the semigroup properties

S(0) = I and S(t+ s) = S(t) S(s), s, t ≥ 0.

In addition, since the solutions are continuous with respect to t and depend continuouslyon the initial data, we conclude that S(t) is a nonlinear C0-semigroup on H. Thereforeproblem (1.8)-(1.12) corresponds to a nonlinear dynamical system (H, S(t)). The existenceof global attractors to this system is discussed in Section 4.

The proof of Theorem 2.1 will be completed trough several lemmas. We study theCauchy problem (2.22) following the arguments of Giorgi and Pata [17], Grasselli andSquassina [20] and Potomkin [29].

We first show two energy estimates that will be used in our analysis. The energy ofthe system (1.8)-(1.12) is defined by

E(t) =1

2∥ut(t)∥22 +

1

2∥∆u(t)∥22 +

1

2∥∇ut(t)∥22 +

1

2M(∥∇u(t)∥22)

+1

2∥θ(t)∥22 +

1

2∥ηt∥2M +

∫Ω

(f(u(t))− hu(t)) dx. (2.27)

6

Lemma 2.2. The energy functional defined in (2.27) satisfies formally

d

dtE(t) = −ω∥∇θ(t)∥22 +

1

2

∫ ∞

0

µ′(s)∥∇ηt(s)∥22 ds, (2.28)

and there exist constants δ0, CE > 0, independent of initial data in H, such that

E(t) ≥ δ0(∥∆u(t)∥22 + ∥∇ut(t)∥22 + ∥θ(t)∥22 + ∥ηt∥2M

)− CE, t ≥ 0. (2.29)

Proof. Let us multiply (1.8) by ut, (1.9) by θ and (1.10) by η. Then we obtain

d

dt

1

2∥ut∥22 +

1

2∥∆u∥22 +

1

2M(∥∇u∥22) +

∫Ω

(f(u)− h) dx

= −ν

∫Ω

∆θutdx,

1

2

d

dt∥θ∥22 = −ω∥∇θ∥22 +

∫ ∞

0

µ(s)

(∫Ω

∆η(s)θ(t)dx

)ds+ ν

∫Ω

∆θutdx,

and1

2

d

dt∥ηt∥M = −

∫ ∞

0

µ(s)

(∫Ω

∆η(s)θ(t)dx

)ds− ⟨ηs, η⟩M.

Taking into account the identity (2.21) we deduce that (2.28) holds.

In order to prove (2.29) let us define

δ0 =1

4

(1− α

λ1− β

λ21

), (2.30)

which is positive because assumption (2.17). Since u ∈ H2(Ω) ∩H10 (Ω), the assumption

(2.13) implies that

M(∥∇u∥22) ≥ − α

λ1∥∆u∥22 − cM ,

and assumption (2.16) implies that∫Ω

f(u) dx ≥ − β

2λ21∥∆u∥22 − cf |Ω|.

Moreover, there exists Cδ0 > 0 such that∫Ω

hu dx ≤ δ0∥∆u∥22 + Cδ0∥h∥22.

Combining above inequalities with (2.27) yields

E(t) ≥ δ0(∥∇ut(t)∥22 + ∥∆u(t)∥22 + ∥θ(t)∥22 + ∥ηt∥2M

)− cM − cf |Ω| − Cδ0∥h∥22.

TakingCE = cM + cf |Ω|+ Cδ0∥h∥22

we obtain estimate (2.29).

7

Lemma 2.3. The operator L defined in (2.23) is the infinitesimal generator of a C0-semigroup in H.

Proof. From energy estimate (2.28), with M = f = h = 0, we see that

⟨Lz(t), z(t)⟩H = −ω∥∇θ(t)∥22 +1

2

∫ ∞

0

µ′(s)∥∇ηt(s)∥22 ds ≤ 0,

for all z(t) ∈ D(L). Then L is a dissipative operator.To show that L is maximal we need to prove that I − L : D(L) → H is onto, which

is equivalent to, given g∗ = (u∗, v∗, θ∗, η∗) ∈ H there exists a solution of z − Lz = g∗ inD(L). In terms of components of z we have

u− v = u∗,v + (I −∆)−1∆2u+ ν(I −∆)−1∆θ = v∗,

θ − ν∆v − ω∆θ −Bµη = θ∗,η − θ + ηs = η∗.

This can be solved following Giorgi and Pata [17] or Potomkin [29].

Then we conclude that operator L ism-dissipative in H. Since D(L) is densely definedin H, the lemma follows from the well-known Lumer-Phillips Theorem (e.g. [28, Thm.1.4.3]).

Lemma 2.4. The operator F defined in (2.25) is locally Lipschitz in H.

Proof. We recall thatI −∆ : H1

0 (Ω) → H−1(Ω)

is a isometrical bijection with respect to the norm ∥u∥2H1

0= ∥∇u∥22 + ∥u∥22 and hence

∥(I −∆)−1w∥H10= ∥w∥H−1 , ∀w ∈ H−1(Ω).

Let us denote z = (u,w1, θ1, η1) and y = (v, w2, θ2, η2). Then we have:

∥F (z)− F (y)∥H ≤ ∥(I −∆)−1[M(∥∇u∥22)∆u−M(∥∇v∥22)∆v − f(u) + f(v) ]∥H10

≤ ∥M(∥∇u∥22)∆u−M(∥∇v∥22)∆v∥H−1 + ∥f(u)− f(v)∥H−1 .

A classical argument for Kirchhoff nonlocal terms shows that∫Ω

(M(∥∇u∥22)∆u−M(∥∇v∥22)∆v

)φdx ≤ C0M∥∇u−∇v∥2 ∥∇φ∥2, ∀φ ∈ H1

0 (Ω), (2.31)

and consequently

∥M(∥∇u∥22)∆u−M(∥∇v∥22)∆v∥H−1 ≤ C0M∥∇u−∇v∥2.

Also, from mean value theorem and assumption (2.15),∫Ω

(f(u)− f(v))φdx ≤ ∥f ′(ξu+ (1− ξ)v)∥r−12r ∥u− v∥2r∥φ∥2

≤ C0f∥∇u−∇v∥2∥φ∥2, (2.32)

8

where 0 ≤ ξ ≤ 1 and C0f > 0 is constant depending on the initial data. Therefore wehave

∥f(u)− f(v)∥H−1 ≤ C0f∥∇u−∇v∥2.Combining above estimates we have

∥F (z)− F (y)∥H ≤ L0∥z − y∥H,

where L0 = max1, C0M , C0f. It follows that F is locally Lipschitz in H.

Proof of Theorem 2.1. Combining the Lemmas 2.3 and 2.4, we have from a classicalresult [28, Thm 6.1.4] that the Cauchy problem (2.22) has a unique local mild solution

z(t) = eLtz0 +

∫ t

0

eL(t−s)F (z(s))ds (2.33)

defined in a maximal interval (0, tmax). In addition, if tmax <∞ then

limt→∞

∥z(t)∥H = +∞. (2.34)

To show that the solution is global, that is, tmax = ∞, let z(t) be a mild solution withinitial data z0 ∈ D(L). From [28, Thm 6.1.5] it is in fact a strong solution and so we canuse energy estimate (2.29) to conclude that

∥z(t)∥2H ≤ 1

δ0(E(0) + CE) , t ≥ 0.

By density this inequality holds for mild solutions. Then clearly (2.34) does not hold andtherefore tmax = ∞. This concludes the proof of item (i) of Theorem 2.1.

The inequality (2.26) is a simple consequence of the representation formula (2.33), thelocal Lipschitz behavior of F and the Gronwall Lemma. Then the continuous dependenceon the initial data for mild solutions also follows. This concludes the proof of item (ii) ofTheorem 2.1.

Finally, as noticed above, from an abstract result [28, Thm 6.1.5] any mild solutionwith initial data in D(L) is strong. This proves the item (iii) of Theorem 2.1.

3 Global attractors

We begin this section by recalling some definitions related to global attractors that canbe found in classical references like [1, 23, 24, 31] or more recent references as Chueshovand Lasiecka [6] and Miranville and Zelik [27].

Let (X,S(t)) be a dynamical system given by a C0-semigroup S(t) on a Banach spaceX. Then a global attractor for S(t) is a compact set A ⊂ X that is fully invariant anduniformly attracting, that is, S(t)A = A for all t ≥ 0 and for every bounded subsetB ⊂ X,

limt→∞

distH(S(t)B,A) = 0,

9

where distH is the Hausdorff semi-distance in X. The fractal dimension of a compact setM ⊂ X is defined by

dimXf M = lim sup

ε→0

lnn(M, ε)

ln(1/ε),

where n(M, ε) is the minimal number of closed balls of radius 2ε which covers M . Givena set S ⊂ X, the unstable manifold emanating from S, denoted by M+(S), is the set ofpoints z ∈ X which belongs to some full trajectory u(t), t ∈ R such that

u(0) = z and limt→−∞

distX(u(t), S) = 0.

Our main result reads as follows.

Theorem 3.1. The dynamical system (H, S(t)) given by Theorem 2.1 has a compactglobal attractor A, with finite fractal-dimension, and characterized by

A = M+(N ),

where N = (u, 0, 0, 0) ∈ H |∆2u −M(∥∇u∥22)∆u + f(u) = h is the set of stationarysolutions.

Remark 3.1. The proof of Theorem 3.1 is presented in the in subsection 3.4. We donot show directly that the system has a bounded absorbing set. Indeed we have foundsome technical difficulty due to the combination of the “extensibility” term M(∥∇u∥22)∆u,the rotational inertia ∆utt and θ. Instead, we will show that the system is gradient andasymptotically compact.

3.1 Some abstract results on dynamical systems

For the readers’s convenience, we present here some abstract results on the existence ofglobal attractors for gradient systems. We also present the concept of quasi-stability.

Let S(t) a strongly continuous semigroup defined in a Banach space H. We say that(H, S(t)) is asymptotically smooth if for any bounded positively invariant set B ⊂ H,there exists a compact set K ⊂ B such that

limt→∞

distH(S(t)B,K) = 0.

We recall that a system (X,S(t)) is called gradient if it possesses a strict Lyapunovfunctional, that is, a functional Φ : X → R such that,

(i) The map t 7→ Φ(S(t)z) is non increasing for each z ∈ X,

(ii) If for some z ∈ X one has Φ(S(t)z) = Φ(z) for all t, then S(t)z = z for all t ≥ 0.

The above condition (ii) means that z must be a stationary point of (X,S(t)).

The following theorem is well-known variation of a classical result in Hale [23, Thm2.4.6]. We consider the one in [7, Cor. 7.5.7].

10

Theorem 3.2. Let us assume that a dynamical system (X,S(t)) is asymptotically smoothand gradient, with Lyapunov functional denoted by Φ. Assume in addition that,

(i) Φ(u) is bounded from above on any bounded subset of X,

(ii) The set ΦR = z |Φ(z) ≤ R is bounded for every R,

(iii) The set of stationary points N is bounded.

Then the system (X,S(t)) has a compact global attractor characterized by A = M+(N ).

Next we recall the concept of quasi-stability stated in Chueshov and Lasiecka [5, 6, 7].In what follows, a seminorm nX(·) defined on a space X is compact if whenever a sequencexj 0 weakly in X one has nX(xj) → 0.

Let X,Y, Z be three reflexive Banach spaces with X compactly embedded into Y andput H = X × Y × Z. Consider the dynamical system (H, S(t)) given by an evolutionoperator

S(t)z = (u(t), ut(t), ξ(t)), z = (u(0), ut(0), ξ(0)) ∈ H, (3.35)

where the functions u and ξ have regularity

u ∈ C(R+;X) ∩ C1(R+;Y ), ξ ∈ C(R+;Z). (3.36)

Then one says that (H, S(t)) is quasi-stable on a set B ⊂ H if there exist a compactsemi-norm nX on X and nonnegative scalar functions a(t) and c(t), locally bounded in[0,∞), and b(t) ∈ L1(R+) with limt→∞ b(t) = 0, such that,

∥S(t)z1 − S(t)z2∥2H ≤ a(t)∥z1 − z2∥2H (3.37)

and

∥S(t)z1 − S(t)z2∥2H ≤ b(t)∥z1 − z2∥2H + c(t) sup0<s<t

[nX(u

1(s)− u2(s))]2, (3.38)

for any z1, z2 ∈ B. The following results are proved in Chueshov and Lasiecka [7, Prop.7.9.4] and [7, Thm. 7.9.6] respectively.

Theorem 3.3. Let (H, S(t)) be a dynamical system given by (3.35) and satisfying (3.36).Then (H, S(t)) is asymptotically smooth if it is quasi-stable on every bounded positivelyinvariant set B of H.

Theorem 3.4. Let (H, S(t)) be a dynamical system given by (3.35) and satisfying (3.36).Suppose that it has a global attractor A. Then if (H, S(t)) is quasi-stable on A, this globalattractor has finite fractal dimension.

11

3.2 Gradient system

It is well-known that the set of stationary points of the Cauchy problem (2.22) is charac-terized by

N = z = (u, 0, 0, 0) ∈ H |L(z) + F (z) = 0 . (3.39)

Moreover, N ⊂ D(L).

Lemma 3.5. The dynamical system (H, S(t)) corresponding to the problem (1.8)-(1.12)is gradient.

Proof. Let us take Φ as the energy E functional defined in (2.27). Then, for z =(u0, u1, θ0, η0), we have from (2.28),

d

dtΦ(S(t)z) = −ω∥∇θ(t)∥22 +

∫ ∞

0

µ′(s)∥∇ηt(s)∥22 ds ≤ 0.

Hence Φ(S(t)z) is non increasing. Now let us suppose Φ(S(t)z) = Φ(z) for all t ≥ 0.Then from above

−ω∥∇θ(t)∥22 +∫ ∞

0

µ′(s)∥∇ηt(s)∥22ds = 0, t ≥ 0.

The both terms in the left-hand side of above identity have the same sign and then∫ ∞

0

µ′(s)∥∇ηt(s)∥22 ds = 0, t ≥ 0.

Letσ∞ = sups |µ(s) > 0.

We note that σ∞ maybe ∞. Then

µ′(s) ≤ −k1µ(s) < 0, s ∈ (0, σ∞),

and thereforeηt(x, s) = 0 for a.e. (x, s) ∈ Ω× (0, σ∞), t ≥ 0.

Hence

∥ηt∥2M =

∫ σ∞

0

µ(s)∥∇ηt(s)∥22 ds+∫ ∞

σ∞

µ(s)∥∇ηt(s)∥22 ds = 0.

Then using equation (1.10) and noting that θ(t) does not depend on s we see that θ(t) = 0a.e. in Ω for all t ≥ 0. Inserting this into equation (1.9) we conclude that ut(t) = 0 a.e. inΩ for all t ≥ 0. This means that u(t) = u0 for all t ≥ 0. So S(t)z = z = (u0, 0, 0, 0).

12

3.3 Quasi-stability

Lemma 3.6. Under the hypotheses of Theorem 3.1, given a bounded set B ⊂ H, letS(t)z1 = (u(t), ut(t), θ(t), η

t) and S(t)z2 = (v(t), vt(t), θ(t), ηt) be two weak solutions of

the problem (1.8)-(1.10) with respective initial conditions z1, z2 in B. Then there existconstants γ, b0 > 0 and CB > 0 depending on B, such that

∥S(t)z1 − S(t)z2∥2H ≤ b0e−γt∥z1 − z2∥2H + CB

∫ t

0

e−γ(t−s)∥∇w(s)∥22ds, (3.40)

where w = u− v.

Proof. Let us write τ = θ − θ and ζ = η − η. Then (w,wt, τ, ζ) is a weak solution of

wtt +∆2w −∆wtt =M(∥∇u∥22)∆u−M(∥∇v∥22)∆v − f(u) + f(v)− ν∆τ, (3.41)

τt = ω∆τ +

∫ ∞

0

µ(s)∆ζt(s)ds+ ν∆wt, (3.42)

ζt = −ζs + τ, (3.43)

with initial conditions

w(0) = u(0)− v(0), wt(0) = ut(0)− vt(0), τ(0) = θ(0)− θ(0), ζ0 = η0 − η0. (3.44)

Then we define the energy functional

G(t) =1

2

∥wt(t)∥22 + ∥∇wt(t)∥22 + ∥∆w(t)∥22 + ∥τ(t)∥22 + ∥ζt∥2M

. (3.45)

Step 1. Given δ1 > 0 there exists a constant Cδ1 > 0, which depends also on B, such that

G′(t) ≤ −ω∥∇τ(t)∥22 +1

2

∫ ∞

0

µ′(s)∥∇ζt(s)∥22 ds+ δ1∥∇wt(t)∥22 + Cδ1∥∇w(t)∥22. (3.46)

To prove this, we multiply (3.41) by wt and (3.42) by τ . Then taking into account thatτ = ζt + ζs we infer that

G′(t) = −ω∥∇τ∥22 +1

2

∫ ∞

0

µ′(s)∥∇ζ(s)∥22 ds+Q1,

where

Q1 =

∫Ω

(M(∥∇u∥22)∆u−M(∥∇v∥22)∆v

)wt dx−

∫Ω

(f(u)− f(v))wt dx.

But from (2.31) and (2.32) with φ = wt we see that for some C(B) > 0, dependent of B,

Q1 ≤ C(B)∥∇w∥2 ∥∇wt∥2.

Then proper application of Young inequality yields (3.46).

13

Step 2. Let us define the functional

ψ(t) =

∫Ω

(wt(t)−∆wt(t))w(t) dx.

Then there exists C1 > 0, depending on B, such that

ψ′(t) ≤ −G(t) + 3

2∥wt(t)∥22 +

3

2∥∇wt(t)∥22 −

1

4∥∆w(t)∥22

+C1∥τ(t)∥22 + C1∥∇w(t)∥22 −1

2

∫ ∞

0

µ′(s)∥∇ζt(s)∥22 ds. (3.47)

Indeed, first we observe that

ψ′(t) =

∫Ω

(wtt(t)−∆wtt(t))w(t) dx+ ∥wt(t)∥22 + ∥∇wt(t)∥22.

From equation (3.41) we see that∫Ω

(wtt −∆wtt)w dx =

∫Ω

(−∆2w − ν∆w)w dx+Q2,

where

Q2 =

∫Ω

(M(∥∇u∥22)∆u−M(∥∇v∥22)∆v

)w dx−

∫Ω

(f(u)− f(v))w dx.

Using (2.31) and (2.32) with φ = w, we infer that for some C(B) > 0, dependent of B,∫Ω

(wtt −∆wtt)w dx ≤ −3

4∥∆w∥22 + ν2∥τ∥22 + C(B)∥∇w∥22.

Then adding −G(t) we get

ψ′(t) ≤ −G(t) + 3

2∥wt∥22 +

3

2∥∇wt∥22 −

1

4∥∆w∥22 +

(ν2 +

1

2

)∥τ∥22

+C(B)∥∇w∥22 +1

2∥η∥2M,

which proves (3.47).

Step 3. Let us define the functional

ϕ(t) =

∫Ω

wt(t)(τ(t) + p(t)) dx where −∆p = τ.

Then given δ2 > 0 there exist constants Cδ2 , C2 > 0 such that

ϕ′(t) ≤ −ν2∥wt∥22 −

ν

2∥∇wt∥22 + δ2∥∆w(t)∥22 + Cδ2∥τ(t)∥22 + C2∥∇w(t)∥22

+ωC2∥∇τ(t)∥22 − C2

∫ ∞

0

µ′(s)∥∇ζt(s)∥22 ds. (3.48)

14

To prove this, we first note that

ϕ′(t) =

∫Ω

wtpt dx+

∫Ω

wtτt dx+

∫Ω

wttp dx+

∫Ω

wttτ dx.

Using equation (3.42) we get∫Ω

wtpt dx =

∫Ω

wt ∆−1

(−ωτ −

∫ ∞

0

µ(s)∆ζ(s) ds− ν∆wt

)dx

≤ −ν2∥wt∥22 + ωC(B)∥∇τ∥22 + C(B)∥ζ∥2M. (3.49)

Analogously, ∫Ω

wtτt dx ≤ −ν2∥∇wt∥22 + ωC(B)∥∇τ∥22 + C(B)∥ζt∥2M. (3.50)

Now in view of equation (3.41),∫Ω

wttp dx =

∫Ω

(−∆2w +∆wtt − ν∆τ

)(−∆−1τ) dx+Q3,

where

Q3 =

∫Ω

(M(∥∇u∥22)∆u−M(∥∇v∥22)∆v

)p dx−

∫Ω

(f(u)− f(v))p dx.

From −∆p = τ we have

∥∇p∥2 ≤1√λ1

∥τ∥2, (3.51)

and using (2.31) and (2.32) with φ = p, we obtain the estimate

Q3 ≤ C(B)∥∇w∥2∥∇p∥2 ≤ C(B)∥∇w∥22 + ∥τ∥22.

Therefore ∫Ω

wttp dx ≤ ∥∆w∥2∥τ∥2 −∫Ω

wttτ dx+ (1 + ν)∥τ∥22 + C(B)∥∇w∥22.

Furthermore, given δ2 > 0 there exists constant Cδ2 > 0 such that∫Ω

wttp dx+

∫Ω

wttτ dx ≤ δ2∥∆w∥22 + Cδ2∥τ∥22 + C(B)∥∇w∥22.

Combining this with (3.49) and (3.50) yields (3.48).

Step 4. Let us define the functional

χ(t) = −∫ ∞

0

µ(s)

(∫Ω

τ(t)ζt(s) dx

)ds.

15

Then given δ3 > 0 there exist constants Cδ3 > 0 such that

χ′(t) ≤ −k02∥τ(t)∥22 + δ3∥∇wt(t)∥22 + ωk0∥∇τ(t)∥22 − Cδ3

∫ ∞

0

µ′(s)∥∇ζt(s)∥22 ds, (3.52)

where k0 =∫µ(s)ds as defined in (2.20). To prove this we first observe that

χ′(t) = −∫ ∞

0

µ(s)

(∫Ω

τt(t)ζt(s)dx

)ds+

∫ ∞

0

µ(s)

(∫Ω

τ(t)ζts(s)dx

)ds− k0∥τ∥22.

Using equation (3.42) we have

−∫ ∞

0µ(s)

(∫Ωτtζ

tdx

)ds = −

∫ ∞

0µ(s)

∫Ωω∆τ ζtdxds−

∫ ∞

0µ(s)

∫Ων∆wt ζ

tdxds

−∫Ω

(∫ ∞

0µ(s)∆ζtds

)(∫ ∞

0µ(s)ζtds

)dx

= I1 + I2 + I3.

We can see that

I1 ≤ ωk0∥∇τ∥22 +ω

4∥ζt∥2M and I3 ≤ k0∥ζt∥2M,

and given δ3 > 0, there exists C(δ3) > 0 such that

I2 ≤ δ3∥∇wt(t)∥22 + C(δ3)∥ζt∥2M.

In addition,∫ ∞

0

µ(s)

(∫Ω

τ(t)ζts(s)dx

)ds =

∫ ∞

0

−µ′(s)

(∫Ω

τ(t)ζt(s)dx

)ds

≤ k02∥τ∥22 −

k′02k0λ1

∫ ∞

0

µ′(s)∥∇ζt∥22 ds,

where k′0 =∫∞0µ′(s)ds. Combining these inequalities we conclude that

χ′(t) ≤ −k02∥τ∥22 + δ3∥∇wt∥22 + ωk0∥∇τ∥22 +

(1

4+ k0 + C(δ3)

)∥ζt∥2M

− k′02k0λ1

∫ ∞

0

µ′(s)∥∇ζt∥22 ds.

Then estimate (3.52) holds with Cδ3 =k′0

2k0λ1+ 1

4+ k0 + C(δ3).

Step 5. Let us define the Lyapunov functional

J (t) = NG(t) + ε1ε2ψ(t) + ε2ϕ(t) + χ(t), (3.53)

where ε1, ε2 ∈ (0, 1) and N > 0 are to be fixed later. Then there exists a β0 > 0 such thatfor N > β0,

β1G(t) ≤ J (t) ≤ β2G(t), ∀ t ≥ 0, (3.54)

16

where β1 = N − β0 and β2 = N + β0. To prove this, let Cs > 0 denote an embeddingconstant. It is easy to see that

|ψ(t)| ≤ ∥wt∥22 + Cs∥∆w∥22 and |χ(t)| ≤ k02∥τ∥22 +

1

2∥ζ∥2M.

Using estimate (3.51) we infer that

|ϕ(t)| ≤ ∥τ∥22 + Cs∥wt∥22.

Then there exists a constant β0 > 0 such that

|ψ(t) + ϕ(t) + χ(t)| ≤ β0G(t),

and therefore (3.54) holds.

Step 6. There exist ε > 0 small and C0 > 0, depending on B, such that

J ′(t) ≤ −εG(t) + C0∥∇w(t)∥22, ∀ t ≥ 0. (3.55)

Indeed, first we chose ε1, δ1 > 0 such that

ε1 <ν

6and δ1 <

ε14.

Then

ε1ψ′(t) + ϕ′(t) ≤ −ε1G(t)−

ν

4∥∇wt(t)∥22 + (C1 + Cδ2)∥τ(t)∥22

+(C1 + C2)∥∇w(t)∥22 −(1

2+ C2

)∫ ∞

0

µ′(s)∥∇ζt(s)∥22 ds.

Next we chose ε2, δ3 > 0 such that

ε2(C1 + Cδ2) <k02

and δ3 <ε2ν

8.

Then

ε2(ε1ψ′(t) + ϕ′(t)) + χ′(t) ≤ −ε1ε2G(t)−

ε2ν

8∥∇wt∥22 + (C1 + C2)∥∇w∥22

+ω(k0 + C2)∥∇τ∥22 −(1

2+ C2 + Cδ3

)∫ ∞

0µ′(s)∥∇ζ(s)∥22 ds.

Taking

N > maxβ0, k0 + C2, 1 + 2C2 + 2Cδ3 and δ1 <ε2ν

8N,

we conclude that

J ′(t) ≤ −ε1ε2G(t) + (C1 + C2 +NCδ1)∥∇w(t)∥22,

and therefore (3.55) holds.

17

Step 7. Conclusion. Combining the right-hand side of (3.54) with (3.55) we obtain

J ′(t) ≤ − ε

β2J (t) + C0∥∇w(t)∥22

and therefore

J (t) ≤ J (0)e− ε

β2t+ C0

∫ t

0

e− ε

β2(t−s)∥∇w(s)∥22 ds.

Using again (3.54) we conclude that

G(t) ≤ β2G(0)e− ε

β2t+ C0

∫ t

0

e− ε

β2(t−s)∥∇w(s)∥22 ds.

Since G(t) is equivalent to the norm ∥(w(t), wt(t), τ(t), ζt)∥2H we conclude that (3.40)

holds.

Lemma 3.7 (Quasi-stability). Under the assumptions of Theorem 3.1 the dynamicalsystem (H, S(t)) is quasi-stable on any bounded positively invariant set B ⊂ H.

Proof. Since (H, S(t)) is defined as the solution operator of (1.8)-(1.12), Theorem 2.1 (i)implies that (3.35) and (3.36) hold with X = V2, Y = V1 and Z = V0 ×M1. In addition,from (2.26) we see that condition (3.37) also holds true. Then we only need to verifystabilization inequality (3.38).

Let B ⊂ H be a bounded set positively invariant with respect to S(t). For z1, z2 ∈ Bwe write S(t)zi = (ui(t), uit(t), θ(t), η

i,t), i = 1, 2. Let us define the seminorm

nX(u) = ∥u∥V1 ,

which is compact in X = V2 since V2 → V1 compactly. Therefore from (3.40) we can write

∥S(t)z1 − S(t)z2∥2H ≤ b(t)∥z1 − z2∥2H + c(t) sup0<s<t

[nX(u

1(s)− u2(s))]2,

where

b(t) = b0e−γt and c(t) = CB

∫ t

0

e−ν(t−s)ds, t ≥ 0.

Therefore the assumptions of quasi-stability on bounded positively invariant sets are ful-filled.

3.4 Proof of the main result

We are going to apply Theorem 3.2. Accordingly, we need to verify that the set ofstationary solutions is bounded.

Lemma 3.8. The set N of stationary solutions of problem (1.8)-(1.12) is bounded in H.

18

Proof. Let u be a stationary solution of (1.8)-(1.12). Then from (3.39) we have

∆2u−M(∥∇u∥22)∆u+ f(u) = h, u ∈ V2,

which by elliptic regularity, u ∈ H4(Ω). Consequently,

∥∆u∥22 +M(∥∇u∥22)∥∇u∥22 +∫Ω

f(u)u dx =

∫Ω

hu dx.

But from the hypotheses (2.13) and (2.16),

M(∥∇u∥22)∥∇u∥22 ≥ − α

λ1∥∆u∥22 − cM and

∫Ω

f(u)u dx ≥ − β

λ21∥∆u∥22 − cf |Ω|.

Then since for ξ > 0 we can write∫Ω

hu dx ≤ ξ∥∆u∥22 +1

4λ21ξ∥h∥22,

it follows that (1− α

λ1− β

λ21− ξ

)∥∆u∥22 ≤ cM + cf |Ω|+

1

4λ21ξ∥h∥22.

Hence fixing ξ < 1− αλ1

− βλ21we conclude the N is bounded.

Proof of Theorem 3.1. Since the system (H, S(t)) corresponds to the solution operatorof problem (1.8)-(1.12) it has the form (3.35) and (3.36). Then from Lemma 3.7 thesystem (H, S(t)) is quasi-stable on bounded positively invariant sets of H and thereforeit is asymptotically smooth by Theorem 3.3.

In view of Lemma 3.5 the system is also gradient with Lyapunov functional Φ = E.Then from definition of the energy we know that Φ(z) is bounded from above on boundedsubsets B of H.

Now, given R > 0, the set

ΦR = U ∈ H|Φ(U) ≤ R

is bounded. Indeed, because of (2.29), we see that for z(t) ∈ ΦR

∥z(t)∥2H ≤ 1

δ0( Φ(z(t)) + CE ) ≤ R + CE

δ0.

We also observe that by Lemma 3.8 the set of stationary solutions are bounded. Thenall the assumptions of Theorem 3.2 are fulfilled and then problem (1.8)-(1.12) has a globalattractor A = M+(N ).

Finally, since a global attractor is itself bounded and positively invariant, Lemma 3.7implies that system (H, S(t)) is quasi-stable on A. Hence Theorem 3.4 implies that Ahas finite fractal dimension.

19

4 Exponential attractors

In this section we are concerned with the existence of exponential attractors, also calledinertial sets, to the problem (1.8)-(1.12). An exponential attractor of a dynamical system(X,S(t)) is a compact set Aexp ⊂ X, with finite fractal dimension, which is positivelyinvariant and attracts exponentially fast the trajectories from any bounded set of initialdata. That is, for any bounded set D ⊂ X, there exist constants tD, CD, γD > 0 such that

distH(S(t)D,Aexp) ≤ CDe−γD(t−tD), t ≥ tD.

The classical theory for exponential attractors is presented in the book by Eden et al [13].See also the survey by Miranville and Zelik [27]. Roughly speaking, for damped hyperbolicequations, in addition to the so called “squeezing property”, we need decompose thesemigroup as S(t) = S1(t) + S2(t), where S(t) is uniformly stable and S2(t) is compact.This approach seems to be very technical to our problem.

In this paper we consider the concept of generalized exponential attractors as presentedin Chueshov and Lasiecka [6, 7]. This differ from the original definition in one assumption,namely, Aexp has finite fractal dimension in an extended phase space H ⊇ H. As we willsee, the existence of (generalized) exponential attractors is greatly simplified for quasi-stable systems.

In order to consider smaller extended spaces H one can use interpolation theory. LetA = −∆ be the Laplacian operator defined in L2(Ω) with domain D(A) = H2

0 (Ω)∩H10 (Ω).

Then we can define the scale of Hilbert spaces Vs = D(As2 ) with inner-space ⟨u, v⟩s =

⟨A s2u,A

s2v⟩0. In particular,

V0 = L2(Ω), V1 = H10 (Ω), V2 = H2(Ω) ∩H1

0 (Ω).

Moreover, we have Vs → Vt whenever s ≥ t. Our main result in this section is thefollowing.

Theorem 4.1. The dynamical system (H, S(t)) given by Theorem 2.1 has a generalizedexponential attractor. More precisely, for any δ ∈ (0, 1], there exists a generalized expo-nential attractor Aexp,δ ⊂ H, with finite fractal dimension in the extended space

H−δ = V2−δ × V1−δ × V−δ ×M1−δ,

where Ms = L2µ(R+;Vs).

To prove this theorem we apply the following abstract result proved in Chueshov andLasiecka [7, Thm. 7.9.9].

Theorem 4.2. Let (H,S(t)) be a dissipative dynamical system satisfying (3.35) and(3.36), and quasi-stable on some bounded absorbing set B. In addition assume there

exists an extended space H ⊇ H such that

∥S(t1)y − S(t2)y∥H ≤ CBT |t1 − t2|γ, t1, t2 ∈ [0, T ], y ∈ B, (4.56)

where CBT > 0 and γ ∈ (0, 1] are constants. Then the dynamical (H, S(t)) has a general-

ized exponential attractor Aexp ⊂ H with finite fractal dimension in H.

20

Proof of Theorem 4.1. Let us take B = z |Φ(z) ≤ R where Φ is the strict Lyapunovfunctional considered in Lemma 3.5. Then for R large it is a positively invariant boundedabsorbing set. This guarantees that the system is quasi-stable on B.

Suppose δ = 1. For strong solutions z(t) with initial data y = z(0) ∈ B we infer from(2.22) and the positive invariance of B that

∥zt(t)∥H−1 ≤ ∥Lz(t)∥H + ∥F (z(t))∥H ≤ CBT , 0 ≤ t ≤ T,

where CBT is a constant. Then it follows that

∥S(t1)y − S(t2)y∥H−1 ≤∫ t2

t1

∥zt(s)∥H−1ds ≤ CBT |t1 − t2|, (4.57)

0 ≤ t1 < t2 ≤ T . This shows that for each y ∈ B, the map t 7→ S(t)y is Holder continuous

in the extended space H = H−1 with exponent γ = 1, so that (4.56) holds. Then theexistence of a generalized exponential attractor whose fractal dimension is finite in H−1

follows from Theorem 4.2.

Now we assume that 0 < δ < 1 and apply the interpolation theorem. Accordingly,there exist constants Dk > 0 such that

∥p∥Vk−δ≤ Dk∥p∥1−δ

Vk∥p∥δVk−1

, ∀ p ∈ Vk, k = 0, 1, 2.

Since we are taking z(t) = (u(t), ut(t), θ(t), ηt) ⊂ B, which is a positively invariant set

contained in some R0-ball, each component of z(t) is bounded by R0. Then

∥u(t)∥V2−δ≤ D2R

1−δ0 ∥u(t)∥δV1

,

and in view of (4.57),∥u(t1)− u(t2)∥2V2−δ

≤ C0|t1 − t2|2δ,for some constant C0 > 0, that depends on B and T . Analogously we get

∥ut(t1)− ut(t2)∥2V1−δ≤ C0|t1 − t2|2δ and ∥θ(t1)− θ(t2)∥2V−δ

≤ C0|t1 − t2|2δ.Furthermore,

∥ηt∥2M1−δ≤ D2

1

∫ ∞

0

µ(s)1−δ∥ηt(s)∥2(1−δ)V1

µ(s)δ∥ηt(s)∥2δV0ds ≤ D2

1R2(1−δ)0 ∥ηt∥2δM0

,

and hence∥ηt1 − ηt2∥2M1−δ

≤ C0|t1 − t2|2δ.Combining above estimates we conclude that

∥S(t1)y − S(t2)y∥H−δ≤ CBT |t1 − t2|δ.

Then for y ∈ B, t 7→ S(t)y is Holder continuous in the extended space H = H−δ withexponent γ = δ. Hence the existence of a generalized exponential attractor, with finitefractal dimension in H−δ, follows from Theorem 4.2.

Acknowledgements The first author was supported by a scholarship from CAPES,Brazilian Ministry of Education. The second author was partially supported by FAPESP,grant 2012/24266-7. They thank Professor J. E. Munoz Rivera for stimulating conversa-tions about thermoelasticity.

21

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Address:

Alisson R. A. Barbosa and T. F. MaInstituto de Ciencias Matematicas e de ComputacaoUniversidade de Sao PauloAvenida Trabalhador Saocalense, 40013566-590 Sao Carlos, Sao Paulo, Brazil

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