3 Global attractors

Long-time dynamics of an extensible plate equation with thermal memory

Alisson Rafael Aguiar Barbosa^∗ To Fu Ma^†

Instituto de Ciˆencias Matem´aticas e de Computa¸c˜ao, Universidade de S˜ao Paulo 13566-590 S˜ao Carlos, S˜ao Paulo, Brazil

Abstract

This work is concerned with long-time dynamics of solutions of extensible plate equations with thermal memory. The problem corresponds to a model of ther- moelasticity based on a theory of non-Fourier heat flux. By considering the case where rotational inertia is present we show that the thermal dissipation is sufficient to stabilize the system and guarantee the existence of a finite-dimensional global attractor. 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 nonlinear problems modeling thermoelastic extensible plates with non-Fourier thermal conduction laws. The model is written as

u_tt+ ∆²u−M(∫

Ω|∇u|²dx)∆u−∆u_tt+f(u) +ν∆θ =h in Ω×R⁺, (1.1) θ_t−ω∆θ−(1−ω)

∫ _∞

0

k(s)∆θ(t−s)ds−ν∆u_t= 0 in Ω×R⁺, (1.2) where Ω is a bounded domain of R² with regular boundary Γ. We assume boundary conditions

u= ∆u= 0 on Γ×R⁺, (1.3)

θ = 0 on Γ×R, (1.4)

and initial conditions

u(x,0) =u₀(x), u_t(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: [email protected].

†Corresponding author. Email: [email protected].

When M is a increasing linear function and ∆u_tt 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 were considered 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) models hyperbolic 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 <1 it 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 of thermoelastic plates of hyperbolic type. Firstly we consider the problem under the absence of rotational inertia −∆u_tt. In Grasselli et al [21], it was proved that the linear system corresponding to (1.1)-(1.5) is not exponentially stable for a large class of kernelsk. They studied (1.1)-(1.5) with M =f = h = 0 = ω = 0 by using semigroup theory. This was also proved, for more general coupling terms, by Coti Zelati et al [9]. With respect to the existence of global attractors, still without rotational inertia, Wu [33] considered the system (1.1)-(1.5) with M = 0. Then by adding a structural damping −∆ut in (1.1) he proved the existence of a global attractor and also the convergence of global solutions to 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 by adding a memory term in the equation (1.1).

On the other hand, Grasselli and Squassina [20] proved, via energy methods, that the presence of rotational inertia −∆ut restores the exponential stability of the linear system corresponding to (1.1)-(1.5). To the authors best knowledge, the existence of global attractors to thermoelastic plates of hyperbolic type was not considered in the presence 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 sole dissipation given by the thermal memory. Then our study essentially complements the one 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. Also extends some results in Giorgi et al [16, 20]. We emphasize that no additional damping in 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 =η^t(x, s) by

η^t(s) =

∫ s 0

θ(t−τ)dτ, (t, s)∈[0,∞)×R⁺. (1.6)

This corresponds to the summed past history of θ and formally satisfies ηt+ηs =θ in Ω, (t, s)∈R⁺×R⁺,

with

η^t(0) = 0 in Ω, t ∈R⁺, and

η⁰(s) =η₀(s) in Ω, s ∈R⁺, where

η₀(s) =

∫ s 0

θ₀(τ)dτ, s ∈R⁺. Now let us write

µ(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)η^t(s)ds, and therefore

(1−ω)

∫ _∞

0

k(s)∆θ(t−s)ds=

∫ _∞

0

µ(s)∆η^t(s)ds. (1.7) From this, problem (1.1)-(1.5) is transformed into the new system

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

∫ _∞

0

µ(s)∆η^t(s)ds−ν∆u_t= 0, (1.9)

η_t+η_s =θ, (1.10)

with boundary conditions

u= ∆u= 0 on Γ×R⁺, θ = 0 on Γ×R⁺, η^t(·,0) = 0 on Γ×R⁺×R⁺, (1.11) and initial conditions

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

u₀(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 hypotheses and prove an existence result for the system (1.8)-(1.12), namely, Theorem 2.1. In Section 3 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. The main result is Theorem 4.1.

2 Assumptions and well-posedness

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

M(s)s≥ −αs−c_M and Mˆ(s)≥ −αs−c_M, ∀s ≥0, (2.13) where ˆM(z) = ∫z

0 M(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 ≥ ¹₂Mˆ(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)| ≤C_f(1 +|s|^r−1), ∀s∈R. (2.15) We also assume there exist constants β ∈[0, λ²₁) and c_f >0 such that

f(s)s ≥ −βs²−c_f and f(s)ˆ ≥ −β

2s²−c_f, ∀s∈R, (2.16) where ˆf(z) = ∫_z

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

lim inf

|s|→∞

f(s)

s >−λ²₁. In addition, we need

α 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 =H₀¹(Ω), V₂ =H²(Ω)∩H₀¹(Ω),

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

M=L²_µ(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 consider its equivalent Cauchy problem. To this end we first remark that derivative η_s can be written in an operator form. Indeed, it is shown in Grasselli and Pata [19] that operator

T η =−η_s, η∈D(T) with

D(T) ={η∈ M |η_s∈ M, η(0) = 0},

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

⟨T η, η⟩_M =

∫ _∞

0

µ^′(s)∥∇η(s)∥²2ds, η∈D(T). (2.21) and the solutions of

η_t=T η+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∆ 0

0 ν∆ ω∆ B_µ

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)

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) with z(0) =z0, for any T >0.

(ii) Ifz₁ andz₂ are two mild solutions of problem(1.8)-(1.12)then there exists a constant c0 >0, depending on initial data, such that

∥z₁(t)−z₂(t)∥_H≤e^c0T∥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), u_t(t), θ(t), η^t),

where (u, u_t, 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 continuously on the initial data, we conclude that S(t) is a nonlinear C⁰-semigroup on H. Therefore problem(1.8)-(1.12)corresponds to a nonlinear dynamical system(H, S(t)). The existence of global attractors to this system is discussed in Section 4.

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

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

E(t) = 1

2∥u_t(t)∥²₂+1

2∥∆u(t)∥²₂+ 1

2∥∇u_t(t)∥²₂+1 2

Mˆ(∥∇u(t)∥²₂) +1

2∥θ(t)∥²2+1

2∥η^t∥²_M+

∫

Ω

( ˆf(u(t))−hu(t))dx. (2.27)

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

dtE(t) = −ω∥∇θ(t)∥²2+1 2

∫ _∞

0

µ^′(s)∥∇η^t(s)∥²2ds, (2.28) and there exist constants δ₀, C_E >0, independent of initial data in H, such that

E(t)≥δ₀(

∥∆u(t)∥²2+∥∇u_t(t)∥²2+∥θ(t)∥²2+∥η^t∥²_M)

−C_E, t ≥0. (2.29)

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

dt {1

2∥u_t∥²2+1

2∥∆u∥²2 +1 2

M(ˆ ∥∇u∥²2) +

∫

Ω

( ˆf(u)−h)dx }

=−ν

∫

Ω

∆θu_tdx, 1

2 d

dt∥θ∥²2 =−ω∥∇θ∥²2+

∫ _∞

0

µ(s) (∫

Ω

∆η(s)θ(t)dx )

ds+ν

∫

Ω

∆θu_tdx,

and 1

2 d

dt∥η^t∥M =−

∫ _∞

0

µ(s) (∫

Ω

∆η(s)θ(t)dx )

ds− ⟨η_s, 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)∩H₀¹(Ω), the assumption (2.13) implies that

M(ˆ ∥∇u∥²2)≥ −α

λ₁∥∆u∥²2−c_M, and assumption (2.16) implies that

∫

Ω

fˆ(u)dx≥ − β

2λ²₁∥∆u∥²2−c_f|Ω|. Moreover, there exists C_δ0 >0 such that

∫

Ω

hu dx≤δ₀∥∆u∥²2+C_δ0∥h∥²2. Combining above inequalities with (2.27) yields

E(t)≥δ₀(

∥∇u_t(t)∥²₂+∥∆u(t)∥²₂+∥θ(t)∥²₂ +∥η^t∥²_M)

−c_M −c_f|Ω| −C_δ0∥h∥²₂. Taking

C_E =c_M +c_f|Ω|+C_δ0∥h∥²2

we obtain estimate (2.29).

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)∥∇η^t(s)∥²₂ds ≤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^∗ in D(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−B_µη = θ^∗, η−θ+ηs = η^∗. This can be solved following Giorgi and Pata [17] or Potomkin [29].

Then we conclude that operatorLism-dissipative inH. SinceD(L) is densely defined in 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 that

I−∆ :H₀¹(Ω) →H⁻¹(Ω) is a isometrical bijection with respect to the norm ∥u∥²_H¹

0 =∥∇u∥²2+∥u∥²2 and hence

∥(I−∆)⁻¹w∥H₀¹ =∥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) ]∥H₀¹

≤ ∥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)

φ dx≤C_0M∥∇u− ∇v∥2∥∇φ∥2, ∀φ∈H₀¹(Ω), (2.31) and consequently

∥M(∥∇u∥²2)∆u−M(∥∇v∥²2)∆v∥H⁻¹ ≤C_0M∥∇u− ∇v∥2. Also, from mean value theorem and assumption (2.15),

∫

Ω

(f(u)−f(v))φ dx ≤ ∥f^′(ξu+ (1−ξ)v)∥^r2r⁻¹∥u−v∥2r∥φ∥2

≤ C_0f∥∇u− ∇v∥2∥φ∥2, (2.32)

where 0 ≤ ξ ≤ 1 and C_0f > 0 is constant depending on the initial data. Therefore we have

∥f(u)−f(v)∥H⁻¹ ≤C_0f∥∇u− ∇v∥2. Combining above estimates we have

∥F(z)−F(y)∥H ≤L₀∥z−y∥H,

where L0 = max{1, C0M, C0f}. It follows thatF is locally Lipschitz in H.

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

z(t) =e^Ltz₀ +

∫ _t

0

e^L(t−s)F(z(s))ds (2.33) defined in a maximal interval (0, t_max). In addition, ift_max<∞ then

tlim→∞∥z(t)∥H = +∞. (2.34) To show that the solution is global, that is, t_max = ∞, let z(t) be a mild solution with initial dataz₀ ∈D(L). From [28, Thm 6.1.5] it is in fact a strong solution and so we can use energy estimate (2.29) to conclude that

∥z(t)∥²_H≤ 1

δ₀(E(0) +C_E), t≥0.

By density this inequality holds for mild solutions. Then clearly (2.34) does not hold and therefore t_max =∞. 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), the local Lipschitz behavior ofF and the Gronwall Lemma. Then the continuous dependence on the initial data for mild solutions also follows. This concludes the proof of item (ii) of Theorem 2.1.

Finally, as noticed above, from an abstract result [28, Thm 6.1.5] any mild solution with 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 can be found in classical references like [1, 23, 24, 31] or more recent references as Chueshov and Lasiecka [6] and Miranville and Zelik [27].

Let (X, S(t)) be a dynamical system given by aC⁰-semigroup S(t) on a Banach space X. Then a global attractor for S(t) is a compact set A ⊂ X that is fully invariant and uniformly attracting, that is, S(t)A = A for all t ≥ 0 and for every bounded subset B ⊂X,

tlim→∞dist_H(S(t)B,A) = 0,

where dist_H is the Hausdorff semi-distance in X. The fractal dimension of a compact set M ⊂X is defined by

dim^X_f 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 coversM. Given a set S ⊂X, the unstable manifold emanating from S, denoted by M+(S), is the set of points z ∈X which belongs to some full trajectory{u(t), t∈R} such that

u(0) =z and lim

t→−∞dist_X(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 compact global 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 stationary solutions.

Remark 3.1. The proof of Theorem 3.1 is presented in the in subsection 3.4. We do not show directly that the system has a bounded absorbing set. Indeed we have found some technical difficulty due to the combination of the “extensibility” term M(∥∇u∥²2)∆u, the rotational inertia ∆u_tt and θ. Instead, we will show that the system is gradient and asymptotically compact.

3.1 Some abstract results on dynamical systems

For the readers’s convenience, we present here some abstract results on the existence of global 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

tlim→∞dist_H(S(t)B, K) = 0.

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

(i) The map t7→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 thatz must be a stationary point of (X, S(t)).

The following theorem is well-known variation of a classical result in Hale [23, Thm 2.4.6]. We consider the one in [7, Cor. 7.5.7].

Theorem 3.2. Let us assume that a dynamical system (X, S(t))is asymptotically smooth and 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 Φ_R={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 seminormn_X(·) defined on a spaceXis compact if whenever a sequence x_j ⇀0 weakly in X one has n_X(x_j)→0.

LetX, Y, Z be three reflexive Banach spaces with X compactly embedded into Y and put H = X ×Y ×Z. Consider the dynamical system (H, S(t)) given by an evolution operator

S(t)z = (u(t), u_t(t), ξ(t)), z = (u(0), u_t(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 compact semi-norm n_X on X and nonnegative scalar functions a(t) and c(t), locally bounded in [0,∞), andb(t)∈L¹(R⁺) with limt→∞b(t) = 0, such that,

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

∥S(t)z¹−S(t)z²∥²_H ≤b(t)∥z¹−z²∥²_H+c(t) sup

0<s<t

[n_X(u¹(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 positively invariant 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 onA, this global attractor has finite fractal dimension.

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)∥∇η^t(s)∥²2ds≤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)∥∇η^t(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)∥∇η^t(s)∥²2ds= 0, t ≥0.

Let

σ_∞= sup{s|µ(s)>0}. We note that σ_∞ maybe ∞. Then

µ^′(s)≤ −k₁µ(s)<0, s ∈(0, σ_∞), and therefore

η^t(x, s) = 0 for a.e. (x, s)∈Ω×(0, σ_∞), t ≥0.

Hence

∥η^t∥²_M =

∫ σ_∞ 0

µ(s)∥∇η^t(s)∥²₂ds+

∫ _∞

σ_∞

µ(s)∥∇η^t(s)∥²₂ds= 0.

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

3.3 Quasi-stability

Lemma 3.6. Under the hypotheses of Theorem 3.1, given a bounded set B ⊂ H, let S(t)z¹ = (u(t), u_t(t), θ(t), η^t) and S(t)z² = (v(t), v_t(t),θ(t),˜ η˜^t) be two weak solutions of the problem (1.8)-(1.10) with respective initial conditions z¹, z² in B. Then there exist constants γ, b₀ >0 and C_B >0 depending on B, such that

∥S(t)z¹−S(t)z²∥²_H ≤ b₀e^−γt∥z¹−z²∥²_H+C_B

∫ _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, w˜ _t, τ, ζ) is a weak solution of w_tt+ ∆²w−∆w_tt =M(∥∇u∥²₂)∆u−M(∥∇v∥²₂)∆v −f(u) +f(v)−ν∆τ, (3.41) τ_t =ω∆τ +

∫ _∞

0

µ(s)∆ζ^t(s)ds+ν∆w_t, (3.42)

ζ_t =−ζ_s+τ, (3.43)

with initial conditions

w(0) =u(0)−v(0), w_t(0) =u_t(0)−v_t(0), τ(0) =θ(0)−θ(0), ζ˜ ⁰ =η⁰−η˜⁰. (3.44) Then we define the energy functional

G(t) = 1 2

{∥w_t(t)∥²2+∥∇w_t(t)∥²2+∥∆w(t)∥²2+∥τ(t)∥²2+∥ζ^t∥²_M}

. (3.45)

Step 1. Givenδ₁ >0 there exists a constantC_δ1 >0, which depends also onB, such that G^′(t)≤ −ω∥∇τ(t)∥²₂ +1

2

∫ _∞

0

µ^′(s)∥∇ζ^t(s)∥²₂ds+δ₁∥∇w_t(t)∥²₂ +C_δ1∥∇w(t)∥²₂. (3.46) To prove this, we multiply (3.41) by w_t and (3.42) by τ. Then taking into account that τ =ζ_t+ζ_s we infer that

G^′(t) =−ω∥∇τ∥²2+ 1 2

∫ _∞

0

µ^′(s)∥∇ζ(s)∥²2ds+Q1, where

Q₁ =

∫

Ω

(M(∥∇u∥²2)∆u−M(∥∇v∥²2)∆v)

w_tdx−

∫

Ω

(f(u)−f(v))w_tdx.

But from (2.31) and (2.32) with φ=w_t we see that for someC(B)>0, dependent of B, Q₁ ≤C(B)∥∇w∥2∥∇w_t∥2.

Then proper application of Young inequality yields (3.46).

Step 2. Let us define the functional

ψ(t) =

∫

Ω

(w_t(t)−∆w_t(t))w(t)dx.

Then there exists C1 >0, depending on B, such that ψ^′(t) ≤ −G(t) + 3

2∥w_t(t)∥²2+ 3

2∥∇w_t(t)∥²2− 1

4∥∆w(t)∥²2

+C₁∥τ(t)∥²2+C₁∥∇w(t)∥²2− 1 2

∫ _∞

0

µ^′(s)∥∇ζ^t(s)∥²2ds. (3.47) Indeed, first we observe that

ψ^′(t) =

∫

Ω

(w_tt(t)−∆w_tt(t))w(t)dx+∥w_t(t)∥²2+∥∇w_t(t)∥²2. From equation (3.41) we see that

∫

Ω

(w_tt−∆w_tt)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 ofB,

∫

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∥w_t∥²2+3

2∥∇w_t∥²2− 1

4∥∆w∥²2+ (

ν² +1 2

)

∥τ∥²2

+C(B)∥∇w∥²2+1 2∥η∥²_M, which proves (3.47).

Step 3. Let us define the functional

ϕ(t) =

∫

Ω

w_t(t)(τ(t) +p(t))dx where −∆p=τ.

Then given δ₂ >0 there exist constants C_δ2, C₂ >0 such that ϕ^′(t) ≤ −ν

2∥w_t∥²2 −ν

2∥∇w_t∥²2+δ₂∥∆w(t)∥²2+C_δ2∥τ(t)∥²2+C₂∥∇w(t)∥²2

+ωC₂∥∇τ(t)∥²2−C₂

∫ _∞

0

µ^′(s)∥∇ζ^t(s)∥²2ds. (3.48)

To prove this, we first note that ϕ^′(t) =

∫

Ω

w_tp_tdx+

∫

Ω

w_tτ_tdx+

∫

Ω

w_ttp dx+

∫

Ω

w_ttτ dx.

Using equation (3.42) we get

∫

Ω

w_tp_tdx =

∫

Ω

w_t∆⁻¹ (

−ωτ −

∫ _∞

0

µ(s)∆ζ(s)ds−ν∆w_t )

dx

≤ −ν

2∥w_t∥²2+ωC(B)∥∇τ∥²2+C(B)∥ζ∥²_M. (3.49) Analogously,

∫

Ω

w_tτ_tdx≤ −ν

2∥∇w_t∥²2+ωC(B)∥∇τ∥²2+C(B)∥ζ^t∥²_M. (3.50) Now in view of equation (3.41),

∫

Ω

w_ttp dx=

∫

Ω

(−∆²w+ ∆w_tt−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

w_ttp dx≤ ∥∆w∥2∥τ∥2−

∫

Ω

w_ttτ dx+ (1 +ν)∥τ∥²₂+C(B)∥∇w∥²₂. Furthermore, given δ₂ >0 there exists constant C_δ2 >0 such that

∫

Ω

w_ttp dx+

∫

Ω

w_ttτ dx≤δ₂∥∆w∥²2+C_δ2∥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)ζ^t(s)dx )

ds.

Then given δ₃ >0 there exist constants C_δ3 >0 such that χ^′(t)≤ −k₀

2∥τ(t)∥²2 +δ₃∥∇w_t(t)∥²2+ωk₀∥∇τ(t)∥²2−C_δ3

∫ _∞

0

µ^′(s)∥∇ζ^t(s)∥²2ds, (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

τ_t(t)ζ^t(s)dx )

ds+

∫ _∞

0

µ(s) (∫

Ω

τ(t)ζ_s^t(s)dx )

ds−k₀∥τ∥²2.

Using equation (3.42) we have

−

∫ _∞

0

µ(s) (∫

Ω

τ_tζ^tdx )

ds = −

∫ _∞

0

µ(s)

∫

Ω

ω∆τ ζ^tdxds−

∫ _∞

0

µ(s)

∫

Ω

ν∆w_tζ^tdxds

−

∫

Ω

(∫ _∞

0

µ(s)∆ζ^tds

) (∫ _∞

0

µ(s)ζ^tds )

dx

= I₁+I₂+I₃.

We can see that

I₁ ≤ωk₀∥∇τ∥²2+ ω

4∥ζ^t∥²_M and I₃ ≤k₀∥ζ^t∥²_M, and given δ₃ >0, there existsC(δ₃)>0 such that

I2 ≤δ3∥∇wt(t)∥²2+C(δ3)∥ζ^t∥²_M. In addition,

∫ _∞

0

µ(s) (∫

Ω

τ(t)ζ_s^t(s)dx )

ds =

∫ _∞

0

−µ^′(s) (∫

Ω

τ(t)ζ^t(s)dx )

ds

≤ k₀

2∥τ∥²₂− k^′₀ 2k₀λ₁

∫ _∞

0

µ^′(s)∥∇ζ^t∥²₂ds, where k₀^′ =∫_∞

0 µ^′(s)ds. Combining these inequalities we conclude that χ^′(t) ≤ −k₀

2∥τ∥²2 +δ₃∥∇w_t∥²2+ωk₀∥∇τ∥²2+ (1

4+k₀+C(δ₃) )

∥ζ^t∥²_M

− k₀^′ 2k₀λ₁

∫ _∞

0

µ^′(s)∥∇ζ^t∥²2ds.

Then estimate (3.52) holds with C_δ3 = _2k^k′0

0λ1 + ¹₄ +k₀ +C(δ₃).

Step 5. Let us define the Lyapunov functional

J(t) =N G(t) +ε₁ε₂ψ(t) +ε₂ϕ(t) +χ(t), (3.53) whereε₁, ε₂ ∈(0,1) andN >0 are to be fixed later. Then there exists aβ₀ >0 such that for N > β₀,

β₁G(t)≤ J(t)≤β₂G(t), ∀t≥0, (3.54)

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

|ψ(t)| ≤ ∥w_t∥²2+C_s∥∆w∥²2 and |χ(t)| ≤ k₀

2∥τ∥²2+1 2∥ζ∥²_M. Using estimate (3.51) we infer that

|ϕ(t)| ≤ ∥τ∥²₂+C_s∥w_t∥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

6 and δ₁ < ε₁ 4. Then

ε₁ψ^′(t) +ϕ^′(t) ≤ −ε₁G(t)−ν

4∥∇w_t(t)∥²2+ (C₁+C_δ2)∥τ(t)∥²2

+(C1+C2)∥∇w(t)∥²2− (1

2+C2

) ∫ _∞

0

µ^′(s)∥∇ζ^t(s)∥²2ds.

Next we chose ε₂, δ₃ >0 such that

ε₂(C₁+C_δ2)< k0

2 and δ₃ < ε2ν 8 . Then

ε₂(ε₁ψ^′(t) +ϕ^′(t)) +χ^′(t) ≤ −ε₁ε₂G(t)−ε₂ν

8 ∥∇w_t∥²₂+ (C₁+C₂)∥∇w∥²₂ +ω(k₀+C₂)∥∇τ∥²₂−

(1

2 +C₂+C_δ3 )∫ _∞

0

µ^′(s)∥∇ζ(s)∥²₂ds.

Taking

N >max{β0, k0+C2,1 + 2C2+ 2Cδ3} and δ1 < ε₂ν 8N, we conclude that

J^′(t)≤ −ε₁ε₂G(t) + (C₁+C₂+N C_δ1)∥∇w(t)∥²2, and therefore (3.55) holds.