Global attractor for the generalized double dispersion equation

Nonlinear Analysis 115 (2015) 103–116

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Global attractor for the generalized double dispersion equation✩ Zhijian Yang a,∗ , Na Feng a , To Fu Ma b a

School of Mathematics and Statistics, Zhengzhou University, No.100, Science Road, Zhengzhou 450001, PR China

b

Institute of Mathematical and Computer Sciences, University of São Paulo, 13566-590 São Carlos, SP, Brazil

article

abstract

info

Article history: Received 1 July 2014 Accepted 10 December 2014 Communicated by Enzo Mitidieri

The paper studies the existence of global attractor for the generalized double dispersion equation arising in elastic waveguide model utt − ∆u − ∆utt + ∆2 u − ∆ut − ∆g (u) = f (x). The main result is concerned with nonlinearities g (u) with supercritical growth. In that case we construct a subclass G of the limit solutions and show that it has a weak global attractor. Especially, in non-supercritical case, the weak topology becomes strong, the further regularity of the global attractor is obtained and the exponential attractor is established in natural energy space. © 2014 Elsevier Ltd. All rights reserved.

Keywords: The generalized double dispersion equation Elastic waveguide model Global weak solutions Non-uniqueness Supercritical exponent Global attractor Exponential attractor

1. Introduction In this paper, we are concerned with the existence of global attractor for the generalized double dispersion equation arising in elastic waveguide model utt − ∆u − ∆utt + ∆2 u − ∆ut − ∆g (u) = f (x) in Ω × R+ ,

(1.1)

where Ω is a bounded domain in R with the smooth boundary ∂ Ω , on which we consider either the hinged boundary condition N

u|∂ Ω = ∆u|∂ Ω = 0,

(1.2)

or the clamped boundary condition u|∂ Ω = 0,

 ∂ u  = 0, ∂ν ∂ Ω

(1.3)

where ν is the unit outward normal on ∂ Ω , and the initial condition u(x, 0) = u0 (x),

ut (x, 0) = u1 (x),

x ∈ Ω,

and the assumptions on g (u) and f will be specified later.

✩ Supported by Natural Science Foundation of China (No. 11271336) and FAPESP (No. 2012/24266-7).

∗

Corresponding author. E-mail addresses: [email protected] (Z. Yang), [email protected] (T.F. Ma).

http://dx.doi.org/10.1016/j.na.2014.12.006 0362-546X/© 2014 Elsevier Ltd. All rights reserved.

(1.4)

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Z. Yang et al. / Nonlinear Analysis 115 (2015) 103–116

In the study of nonlinear wave propagation in elastic waveguide, on account of the energy exchange between the waveguide and the external medium through the lateral surfaces of the waveguide, Samsonov et al. [28,27] established the so-called cubic double dispersion equation 1

(cu3 + 6u2 + autt − buxx + dut )xx (1.5) 4 to describe the longitudinal displacement of the elastic rod. Here a, b, c > 0 and d ̸= 0 are some constants depending on the Young modulus, the shearing modulus, density of the waveguide and the Poisson coefficient. Obviously, Eq. (1.1) includes (1.5) as its special case. There have been lots of research studies on the well-posedness, blowup, asymptotic behavior and other properties of solutions for both the IBVP and the IVP of the equation of type (1.1) (see [1,5,6,8,9,22–26,30–33] and references therein). While for the investigation on the global attractor to Eq. (1.1), one can see [14,15,34–36] and references therein. Global attractor is a basic concept in the research studies of the asymptotic behavior of the dissipative system. From the physical point of view, the global attractor of the dissipative equation (1.1) represents the permanent regime that can be observed when the excitation starts from any point in natural energy space, and its dimension represents the number of degree of freedom of the related turbulent phenomenon and thus the level of complexity concerning the flow. All the information concerning the attractor and its dimension from the qualitative nature to the quantitative nature then yield valuable information concerning the flows that this physical system can generate. On the physical and numerical sides, this dimension gives one an idea of the number of parameters and the size of the computations needed in numerical simulations. However, the global attractor may possess an essential drawback, namely, the rate of attraction may be arbitrarily slow and it cannot be estimated in terms of physical parameters of the system under consideration. While the exponential attractor overcomes the drawback because not only it has finite fractal dimension but also its contractive rate is exponential and measurable in terms of the physical parameters. Chueshov and Lasiecka [12,11] studied the longtime behavior of solutions to the Kirchhoff–Boussinesq plate equation utt − uxx =

utt + kut + ∆2 u = div[f0 (∇ u)] + ∆[f1 (u)] − f2 (u)

(1.6)

with Ω ⊂ R and the clamped boundary condition (1.3). Here k > 0 is the damping parameter, the mapping f0 : R → R2 and the smooth functions f1 and f2 represent (nonlinear) feedback forces acting upon the plate, in particular, 2

f0 (∇ u) = |∇ u|2 ∇ u,

2

f1 (u) = u2 + u.

Ignoring both restoring force f0 (∇ u) and feedback force f2 (u) and replacing the inertial term utt by ϵ utt , with ϵ > 0 (the relaxation time) sufficiently small, Eq. (1.6) becomes the modified Cahn–Hilliard equation

ϵ utt + ut − ∆(−∆u + f (u)) = g ,

(1.7)

which is proposed by Galenko et al. [16–18] to model rapid spinodal decomposition in non-equilibrium phase separation processes. Grasselli et al. [20,19,21] studied the well-posedness and the longtime dynamics of Eq. (1.7) in both 2D and 3D cases, with hinged boundary condition. They established the existence of the global and exponential attractor for ϵ = 1 in 2D case, and for ϵ > 0 sufficiently small in 3D case. Taking ϵ = 1 in (1.7) or taking f0 (∇ u) = ∇ u, f2 = 0 in (1.6), and taking into account the inertial force represented by −∆utt and replacing the weak damping ut by a strong one −∆ut , Eq. (1.1) arises. In 1D case, Dai and Guo [14,15] established in phase space E2 = H 2 ∩ H01 × H01 the finite dimensional global attractor for the IBVP of Eq. (1.1), with hinged boundary condition (1.2). For the multidimensional case, Yang [34] established in E2 the global attractor provided that the growth exponent p of the nonlinearity g (u) is subcritical, that is, 1 ≤ p < (N −N2)+ , with N ≤ 5, where and in the context a+ = max{a, 0}. Under the similar assumptions the author [35] also discussed the existence of global attractor for Eq. (1.1) on RN . Here the growth exponent p˜ = NN−2 (N ≥ 3) is called critical because one cannot get the uniqueness of weak solutions and cannot define the solution semigroup according to the traditional manner as p > p˜ . When p > p˜ , by introducing the trajectory dynamical system, which does not require the uniqueness of solutions and is developed by Chepyzhov and Vishik [10], Yang [36] established the so-called trajectory attractor but in the trajectory phase space equipped with weak∗ topology and not in natural energy space. In order to establish the global attractor in the sense of strong topology in the case of supercritical nonlinearity and without the uniqueness of solutions, Ball [2,3] proposed the concept of generalized semiflows and use it (see [3]) to study the longtime dynamics of the semi-linear evolution equation utt − ∆u + β ut + g (u) = 0,

(1.8)

on a bounded domain Ω ⊂ RN with Dirichlet boundary condition. In the supercritical nonlinearity case:

|g (s)| ≤ C (1 + |s|r ),

(1.9)

based on an unproved assumption that every weak solution satisfies the energy equation, he showed with r > that the related generalized semiflow possesses in natural energy space a global attractor without the requirement for the uniqueness of weak solutions. But by now, to the best of our knowledge, the unproved assumption is still an open problem. Recently, Carvalho, Cholewa and Dlotko [7] proposed the concept of ‘the subclass LS of the limit solutions’ and proved that the corresponding subclass LS of Eq. (1.8) has a weak global attractor provided that 1 < r < (NN−+22)+ . N , (N −2)+

Z. Yang et al. / Nonlinear Analysis 115 (2015) 103–116

105

Motivated by the idea in [7], by introducing the limit solution semigroup in a new manner, we construct ‘the subclass G of the limit solutions’ for both problem (1.1), (1.2), (1.4) and problem (1.1), (1.3), (1.4) and prove in a simplified method that G has a weak global attractor in natural energy space E1 provided that 1 ≤ p < p∗ ≡ (NN−+22)+ . Especially, in non-supercritical

case, that is, 1 ≤ p ≤ p˜ = (N −N2)+ , the weak topology becomes strong, the E2 regularity of the global attractor is obtained and the exponential attractor is established in E1 . In comparison with the results in [34,35], the contribution of the paper lies in that:

1. The weak global attractor is established in natural energy space E1 in supercritical case. See Theorem 3.4. 2. The critical case p = p˜ is solved in energy space E1 . In concrete, when 1 ≤ p ≤ p˜ , the global and exponential attractor in E1 is established, and the E2 -regularity of the global attractor is obtained. See Theorem 4.2. 3. The restriction N ≤ 5 is removed in subcritical case. See Theorem 4.2. The plan of the paper is as follows. In Section 2, the global existence of the weak solutions is discussed by virtue of the viscous elimination method and the energy method. In Section 3, the subclass G of the limit solutions is constructed and the existence of weak global attractor is established. In Section 4, the weak topology becomes strong and the exponential attractor is established for non-supercritical case. 2. Global existence of weak solutions For brevity, we use the following abbreviations: Lp = Lp (Ω ),

H k = H k (Ω ),

H = L2 ,

∥ · ∥ = ∥ · ∥L2 ,

∥ · ∥ p = ∥ · ∥ Lp ,

with p ≥ 1, and

 V2 =

H 2 ∩ H01 for the hinged boundary condition (1.2), H02 for the clamped boundary condition (1.3),

where H k are the L2 -based Sobolev spaces and H0k are the completion of C0∞ (Ω ) in H k for k > 0, ϕu = (u, ut ). The notation (·, ·) for the H-inner product will also be used for the notation of duality pairing between dual spaces, C (· · ·) denotes positive constants depending on the quantities appearing in parenthesis. For the hinged boundary condition (1.2), we define the operator A : V2 → V2′ (the dual space of V2 = H 2 ∩ H01 ),

(Au, v) = (∆u, ∆v) for any u, v ∈ V2 . s

Then, the operators As (s ∈ R) are strictly positive and the spaces Vs = D(A 4 ) are Hilbert spaces with the scalar products and the norms s

s

s

(u, v)s = (A 4 u, A 4 v),

∥u∥Vs = ∥A 4 u∥,

respectively. Obviously, 1

1

∥u∥V2 = ∥A 2 u∥ = ∥∆u∥,

∥u∥V1 = ∥A 4 u∥ = ∥∇ u∥.

1 Rewriting Eq. (1.1) in the operator equation and applying A− 2 to the resulting expression, we get the Cauchy problem equivalent to problem (1.1), (1.2), (1.4):



1





1

A− 2 + I utt + I + A 2

u(0) = u0 ,



1 u + ut + g (u) = A− 2 f ,

ut (0) = u1 .

(2.1) (2.2)

For the clamped boundary condition (1.3), we define the operator Λ = −∆, with Dirichlet zero boundary condition. Obviously, Λ : H 2 ∩ H01 → H is a strict positive and self-adjoint operator and an isomorphism. Define the operator ˜ = Λ|H 2 : H02 → H (the restriction of Λ on H02 ). Obviously, for any u, v ∈ H02 , Λ 0

˜ u, v) = (∇ u, ∇v), (Λ

˜ u, u) = ∥∇ u∥2 ≥ 0. (Λ

˜ is also a strict positive and self-adjoint operator and an isomorphism. So, we can define the power Λ ˜ s , s ∈ R, and That is, Λ ˜ s/2 ) = {u ∈ H |Λ ˜ s/2 u ∈ H } for s ≥ 0, Vs = D(Λ ˜ s/2 ) = (D(Λ ˜ −s/2 ))′ for s < 0. The spaces Vs = D(Λ ˜ s/2 ) the spaces Vs = D(Λ are Hilbert spaces with the scalar products and the norms ˜ s/2 u, Λ ˜ s/2 v), (u, v)s = (Λ

˜ s/2 u∥, ∥u∥Vs = ∥Λ

˜ ) = H02 , respectively. In particular, V2 = D(Λ ˜ u∥ = ∥∆u∥, ∥u∥V2 = ∥Λ

˜ 1/2 u∥ = ∥∇ u∥. ∥u∥V1 = ∥Λ

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Z. Yang et al. / Nonlinear Analysis 115 (2015) 103–116

˜ is also an isomorphism from H to V−2 = (H02 )′ , rewriting Eq. (1.1) in the operator equation and Noticing that the mapping Λ − 1 ˜ , we get the Cauchy problem equivalent to problem (1.1), (1.3), (1.4): applying Λ ˜ −1 + I )utt + (I + Λ ˜ )u + ut + g (u) = Λ ˜ −1 f , (Λ u(0) = u0 , ut (0) = u1 .

(2.3) (2.4)

All in all, for each s ∈ R,

 Vs =

D(As/4 ) for the hinged boundary condition (1.2), ˜ s/2 ) for the clamped boundary condition (1.3). D(Λ

We denote the Banach space Es ≡ Vs × Vs−1 for each s ∈ R, which is equipped with the usual graph norm, for example, E1 = V1 × H, and

∥(u, v)∥2E1 = ∥u∥2V1 + ∥v∥2 . ˜ 2 one can see that the similar For clarity, we concentrate the argument on problem (2.1)–(2.2) because by replacing A by Λ arguments and conclusions hold true for problem (2.3)–(2.4). We first consider the parabolic type perturbations of Eq. (2.1): 

1





1

A− 2 + I utt + I + A 2



1 1 u + 2ηA 4 ut + ut + g (u) = A− 2 f ,

(2.5)

with parameter η > 0. Theorem 2.1. Assume that (H1 ) g ∈ C 1 (R), lim inf |s|→∞

g ( s) s

 > − λ1 ,

(2.6)

where λ1 (> 0) is the first eigenvalue of the operator A, and when N ≥ 2,

|g ′ (s)| ≤ C (1 + |s|p−1 ),

s ∈ R,

2 as N ≥ 3. where 1 ≤ p < ∞ as N = 2; 1 ≤ p ≤ p∗ ≡ NN + −2 (H2 ) (u0 , u1 ) ∈ E1 , f ∈ V−3 . Then problem (2.5), (2.2) admits a unique weak solution u, with ϕu = (u, ut ) ∈ L∞ (R+ , E1 ) ∩ Cw (R+ , E1 ). Moreover, when 1 ≤ p < p∗ , the solution is Lipschitz continuous in the weaker space E1/2 , that is,

∥(z (t ), zt (t ))∥2E1/2 ≤ Cekt ∥(z (0), zt (0))∥2E1/2 ,

t ≥ 0,

(2.7)

for some C , k > 0, where z = u − v, u and v are respectively the weak solutions of Eq. (2.5) corresponding to initial data (u0 , u1 ) and (v0 , v1 ). Remark 2.1.

(i) The formula (2.6) implies that there exists a positive constant θ : 0 <

√ λ1 − θ ≪ 1 such that

(g (u), u) ≥ −θ ∥u∥2 − C ,  θ θ 1 G(u)dx ≥ − ∥u∥2 − C ≥ − √ ∥A 4 u∥2 − C , 2 2 λ1 Ω u where G(u) = 0 g (τ )dτ . (ii) When N = 1, g (s) = 4c s3 + 32 s2 , Eq. (1.1) becomes the cubic double dispersion equation (1.5), with a = b = d = 4. Obviously, g ∈ C 1 (R), and g (s) lim inf = +∞. |s|→∞

s

That is, g satisfies the assumptions of Theorem 2.1. (iii) One can see from the proof of Theorem 2.1 that the parameter η > 0 is indispensable to guarantee the uniqueness of the weak solution as p˜ < p < p∗ . (iv) The conclusions of Theorem 2.1 are also valid for the parabolic type perturbations of Eq. (2.3): 1

˜ −1 + I )utt + (I + Λ ˜ )u + 2ηΛ ˜ 2 ut + ut + g ( u) = Λ ˜ −1 f (Λ ˜ 2. provided that λ1 in (2.6) is the first eigenvalue of Λ

(2.8)

Z. Yang et al. / Nonlinear Analysis 115 (2015) 103–116

107

˜ 2 for the clamped boundary condition), with Under the assumptions of Theorem 2.1 (where λ1 is the first eigenvalue of Λ ∗ 1 ≤ p < p , we can define the solution operator Tη (t ) : E1 → E1 (with η > 0), Tη (t )ϕ0 = ϕ(t ) = (u(t ), ut (t ))

for every ϕ0 ∈ E1 ,

t ≥ 0,

where u is the weak solution of problem (2.5), (2.2) (problem (2.8), (2.2)), with ϕ(0) = ϕ0 . Theorem 2.1 shows that {Tη (t )} constitutes a semigroup on E1 , which is Lipschitz continuous in the topology of E1/2 . Theorem 2.2 (Existence of the Limit Solutions). Let the assumptions of Theorem 2.1 be in force (where λ1 is the first eigenvalue of ˜ 2 for the clamped boundary condition), with 1 ≤ p < p∗ , and let ϕ0 ∈ E1 , ηn → 0+ , ϕ0n ⇀ ϕ0 . Then there exists subsequence Λ {nk } such that n

Tηnk (t )ϕ0 k ⇀ ϕ(t ) = (u(t ), ut (t )),

(2.9)

where u is the weak solution of problem (2.1)–(2.2) (problem (2.3), (2.4)), and the sign ‘‘ ⇀’’ denotes ‘‘weakly convergent in E1 ’’. Lemma 2.3 ([29]). Let X , B and Y be the Banach spaces, X ↩→↩→ B ↩→ Y , W = {u ∈ Lp (0, T ; X )|ut ∈ L1 (0, T ; Y )},

with 1 ≤ p < ∞,

W1 = {u ∈ L∞ (0, T ; X )|ut ∈ Lr (0, T ; Y )},

with r > 1.

Then, W ↩→↩→ Lp (0, T ; B),

W1 ↩→↩→ C ([0, T ]; B).

Proof of Theorem 2.1. We first obtain a priori estimate to the solutions of problem (2.5), (2.2). Using the multiplier ut in (2.5), we have d dt

1

H (u, ut ) + 2η∥A 8 ut ∥2 + ∥ut ∥2 = 0,

t > 0,

(2.10)

where 1

H ( u, ut ) =

2 1

∥A

− 41

2

1 4

2

2

ut ∥ + ∥ ut ∥ + ∥ A u∥ + ∥ u∥

2





1

+ Ω

G(u)dx − (A− 2 f , u)

θ  1 2 ∥ ut ∥ + 1 − √ ≥ ∥A 4 u∥ − C (∥f ∥V−3 ). 4 λ1 

2

Hence,

∥ϕu (t )∥ + 2 E1

 t



1

2η∥A 8 ut (τ )∥2 + ∥ut (τ )∥2 dτ ≤ C0 ,

t ≥ 0,

(2.11)

0

where C0 = (∥(u0 , u1 )∥E1 , ∥f ∥V−3 ). For any φ ∈ V−1 , 1

∥(I + A− 2 )φ∥V−1 ≤ C ∥φ∥V−1 ,



1

I + A− 2



 φ, φ ≥ ∥φ∥2V−1 ,

1

i.e. the operator I + A− 2 is bounded and strictly positive on V−1 . Hence, 1

∥(I + A− 2 )−1 ∥L(V−1 ) ≤ C , and thus 1

∥utt (t )∥V−1 ≤ ∥(I + A− 2 )−1 ∥L(V−1 ) [∥f ∥V−3 + ∥u∥V1 + ∥ut ∥ + ∥g (u)∥V−1 ] ≤ C0 ,

t ≥ 0,

(2.12)

where we have used Eq. (2.5) and the fact

∥g (u)∥V−1 ≤ C ∥g (u)∥1+1/p ≤ C (1 + ∥u∥pp+1 ) ≤ C0 . Now, we look for the approximate solutions un of problem (2.5), (2.2) of the form un (t ) =

n 

Tjn (t )wj ,

j =1

where Awj = λj wj , j = 1, 2, . . . , {wj } is an orthonormal basis in H, and at the same time an orthogonal one in V2 , and Tjn (t ) = (un , wj ) with

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Z. Yang et al. / Nonlinear Analysis 115 (2015) 103–116







1

A− 2 + I untt , wj +

 1  = A− 2 f , wj , (u (0), n

unt



 1

I + A2





1



un , wj + 2η A 4 unt , wj + (unt , wj ) + (g (un ), wj )

t > 0, j = 1, . . . , n,

(2.13)

(0)) = (u0n , u1n ) → (u0 , u1 ) in E1 .

Obviously, the estimates (2.11)–(2.12) are valid for un . So we can extract a subsequence, still denoted by {un }, such that + (un , unt ) → (u, ut ) weakly∗ in L∞ loc (R ; E1 ); + weakly∗ in L∞ loc (R ; V−1 );

untt → utt

(u (t ), n

unt

(2.14)

(t )) ⇀ (u(t ), ut (t )) for t ≥ 0.

Applying Lemma 2.3 to (2.14), we have

(un , unt ) → (u, ut ) in C ([0, T ]; V1−δ × V−δ )

(2.15)

for δ : 0 < δ ≪ 1, and hence un → u

a.e. in Ω ,

g (un ) → g (u) weakly in L1+1/p

(2.16)

for t ≥ 0. Letting n → ∞ in (2.13) we see that the limit function u is a weak solution of problem (2.5), (2.2), with ϕu = (u, ut ) ∈ L∞ (R+ ; E1 ) ∩ Cw (R+ ; E1 ). 1 Obviously, by now the viscous perturbation 2ηA 4 ut does not play role, that is, the convergence (2.14)–(2.16) and the existence of the weak solutions hold true for η ≥ 0. Now, we show that ϕu = (u, ut ) is Lipschitz continuous in the weak space E1/2 . In fact, let u, v be two solutions of problem (2.5), (2.2) as shown above corresponding to initial data u0 , u1 and v0 , v1 , respectively. Then z = u − v solves





1



1

A− 2 + I ztt + I + A 2



z (0) = u0 − v0 ≡ z0 , − 41

Using the multiplier A d dt

1

z + 2ηA 4 zt + zt + g (u) − g (v) = 0,

(2.17)

zt (0) = u1 − v1 ≡ z1 .

zt + ϵ z in (2.17), we get 1

1

1



1



H1 (z , zt ) + ∥A− 8 zt ∥2 + (2η − ϵ)∥zt ∥2 − ϵ∥A− 4 zt ∥2 + ϵ∥z ∥2 + ϵ∥A 4 z ∥2 = − g (u) − g (v), A− 4 zt + ϵ z , (2.18)

where H1 (z , zt ) =

1

 3 1 1 1 ∥A− 8 zt ∥2 + ∥A− 8 zt ∥2 + ∥A− 8 z ∥2 + ∥A 8 z ∥2 2  1   1 1 1 + ϵ A− 4 zt , A− 4 z + (zt , z ) + ∥z ∥2 + η∥A 8 z ∥2 2

− 18

1 8

v ∥A z ∥2 + ∥ A

zt ∥2 = ∥z ∥2V1/2 + ∥zt ∥2V−1/2

(2.19)

for ϵ > 0 suitably small, and where we have used the estimate

ϵ



1

1

A− 4 z , A− 4 zt



 + (z , zt ) ≤ ϵ(∥z ∥2V1/2 + ∥zt ∥2V−1/2 ).

On account of V1−δ ↩→ Lp+1 for δ : 0 < δ ≪ 1, and by virtue of the interpolation theorem we have the control for the right hand side of (2.18) 1 p−1 2 2 |(g (u) − g (v), z )| ≤ C (1 + ∥u∥pp− +1 + ∥v∥p+1 )∥z ∥p+1 ≤ C ∥z ∥V1−δ ≤ ϵ∥z ∥2V1 + C (ϵ)∥z ∥2 ,

|(g (u) − g (v), A

− 14

p−1

p−1

(2.20)

− 14

zt )| ≤ C (1 + ∥u∥p+1 + ∥v∥p+1 )∥z ∥p+1 ∥A

zt ∥p+1 δ δ 1−δ ≤ C ∥z ∥V1−δ ∥zt ∥V−δ ≤ C ∥z ∥V1−δ ∥ z ∥ ∥ z ∥ t V−1 ∥zt ∥ 1 ≤ ϵ 2 ∥z ∥2V1 + ϵ∥zt ∥2 + C (ϵ)(∥z ∥2 + ∥zt ∥2V−1 ).

(2.21)

Inserting (2.20)–(2.21) into (2.18) and making use of (2.19) we get d dt

H1 (z , zt ) ≤ CH1 (z , zt )

(2.22)

for ϵ : 0 < ϵ < η suitably small. Applying the Gronwall inequality to (2.22), we get the desired Lipschitz continuity (2.7). Theorem 2.1 is proved.

Z. Yang et al. / Nonlinear Analysis 115 (2015) 103–116

109

Proof of Theorem 2.2. Let

ϕ n (t ) = (un (t ), unt (t )) = Tηn ϕ0n ,

with ηn > 0, ηn → 0+ , ϕ0n ∈ E1 .

Then un solves Eq. (2.5), with ϕ n (0) = ϕ0n , that is, for any φ ∈ V1 ,





1



1

A− 2 + I untt + I + A 2



1

1

un + 2ηn A 4 unt + unt + g (un ) − A− 2 f , φ



= 0,

t > 0.

(2.23)

Obviously, by the lower semi-continuity of the norm of the weak limit and the uniqueness of the weak solutions, the estimates (2.11)–(2.12) are still valid for un , so one can extract a subsequence, still denoted by un , such that (2.14)–(2.16) hold true. On account of



1

2ηn A 4 unt , φ



  1 = 2ηn unt , A 4 φ → 0,

letting n → ∞ in (2.23) we see that the limit function u is the weak solution of problem (2.1)–(2.2). This completes the proof. 3. Weak global attractor Let 2E1 be the space of all subsets of E1 . Define the operator T (t ) : 2E1 → 2E1 , n

T (t )ϕ0 = ϕ(t ) = (w) lim Tηnk (t )ϕ0 k , k→∞

t ≥0

(3.1) n

where and in the context the sign ‘‘(w) lim’’ denotes the weak limit in the phase space E1 and {ϕ0 k } is as shown in (2.9). By Theorem 2.2, T (t ) is well defined, ϕ(t ) = T (t )ϕ0 and ϕ(t ) may be multiple-valued (if without the uniqueness of the weak solutions) because there may be lots of subsequences of {ϕ0n } such that the weak limit on the right hand side of (3.1) exists and all these weak limits are denoted by ϕ(t ). Obviously, for any subset B ⊂ E1 , T (t )B = {ϕ(t ) = T (t )ϕ0 |ϕ0 ∈ B}.

(3.2)

Theorem 2.2 shows that for every ϕ0 ∈ E1 , problem (2.1)–(2.2) (problem (2.3)–(2.4)) possesses at least one ‘‘limit solution’’. Let G be the set of all these ‘‘limit solutions’’, that is,

G = {ϕ ∈ L∞ (R+ ; E1 )|ϕ(t ) = T (t )ϕ0 , ϕ0 ∈ E1 }. Lemma 3.1. The subclass G of the limit solutions is of the following properties: (i) (Existence) For each ϕ0 ∈ E1 , there exists at least one ϕ ∈ G with ϕ ∈ Cw (R+ ; E1 ), ϕ(0) = ϕ0 . (ii) (Translates of solutions are solutions) If ϕ ∈ G and τ ≥ 0, then ϕ τ ∈ G, where ϕ τ (t ) = ϕ(t + τ ), t ∈ [0, ∞). (iii) (Concatenation) If ϕ, ψ ∈ G, with ψ(0) = ϕ(t ), t ≥ 0, then θ ∈ G, where

θ (τ ) :=



ϕ(τ ), ψ(τ − t ),

0 ≤ τ ≤ t, t < τ.

(3.3)

Proof. (i) The property (i) follows directly from Theorem 2.2. (ii) For any ϕ ∈ G and τ ≥ 0,

ϕ τ (t ) = ϕ(t + τ ) = T (t + τ )ϕ(0) = (w) lim Tηnk (t + τ )ϕ0 k n

k→∞

n

= (w) lim Tηnk (t )Tηnk (τ )ϕ0 k = T (t )ϕ(τ )(= T (t )T (τ )ϕ(0)), k→∞

that is, ϕ τ ∈ G. Moreover, we see from (3.1) and (3.4) that T (0)ϕ0 = ϕ0 ,

T (t + τ )ϕ0 = T (t )T (τ )ϕ0

for any ϕ0 ∈ E1 and t , τ ≥ 0.

Hence, for any subset B ⊂ E1 , T (0)B = B,

T (t + τ )B = {ϕ(t + τ ) = T (t )T (τ )ϕ0 |ϕ0 ∈ B} = T (t )T (τ )B

for t , τ ≥ 0. That is, T (0) = I ,

T (t + τ ) = T (t )T (τ )

for t , τ ≥ 0,

T (t ) constitutes a semigroup on 2 . (iii) If ϕ, ψ ∈ G, with ψ(0) = ϕ(t ), t ≥ 0, then when τ ≥ t, E1

ψ(τ − t ) = T (τ − t )ψ0 = T (τ − t )ϕ(t ) = T (τ − t )T (t )ϕ(0) = T (τ )ϕ(0) = ϕ(τ ), so θ ∈ G, where θ (τ ) is as shown in (3.3).

(3.4)

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Z. Yang et al. / Nonlinear Analysis 115 (2015) 103–116

Remark 3.1. Comparing the definition of the subclass G of the limit solutions here with that of the generalized semiflows G1 in [3] one can easily find that G1 (if it exists) is a true subset of G because G does not satisfy the fourth property of G1 : ‘‘the upper-semicontinuity with respect to initial data’’ (cf. [3]). Definition 3.2. The set A ⊂ E1 is called the weak (E1 , Es ) attractor of the subclass G of the limit solutions, with 0 ≤ s < 1, if the following conditions are satisfied: (i) A is a bounded closed set in E1 and a compact set in Es ; (ii) A is an invariant set, i.e, T (t )A = A, t ≥ 0; (iii) A attracts in the topology of Es the bounded set in E1 , that is, for any bounded set B ⊂ E1 distEs {T (t )B, A} → 0

as t → ∞.

Lemma 3.3 (Gronwall-type Lemma [4]). Let X be a Banach space, and let Z ⊂ C (R+ , X ). Let Φ : X → R be a function such that sup Φ (z (t )) ≥ −η,

Φ (z (0)) ≤ K ,

t ∈R+

for some η, K ≥ 0 and every z ∈ Z. In addition, assume that for every z ∈ Z the function t → Φ (z (t )) is continuously differentiable, and satisfies the differential inequality d dt

Φ (z (t )) + δ∥z (t )∥2 ≤ k η+K

for some δ > 0 and k ≥ 0 independent of z ∈ Z. Then, for every γ > 0 there is t0 = γ

Φ (z (t )) ≤ sup{Φ (ζ ) : δ∥ζ ∥ ≤ k + γ }, 2

ζ ∈X

> 0 such that

t ≥ t0 .

˜ 2 for the clamped boundary Theorem 3.4. Let the assumptions of Theorem 2.1 be in force (where λ1 is the first eigenvalue of Λ condition), with 1 ≤ p < p∗ . Then the subclass G of the limit solutions has a (E1 , Es ) (0 ≤ s < 1) attractor. Proof. Existence of the bounded absorbing set. We first formally get a priori estimate for the solution of problem (2.5), (2.2). Using the multiplier u in (2.5), we have



d dt

1

1

A− 4 u, A− 4 ut



− 12

= A





 1 1 1 + (u, ut ) + ∥u∥2 + η∥A 8 u∥2 ∥A 4 u∥2 + ∥u∥2 + (g (u), u) 2

− 14

f , u + ∥A

ut ∥ 2 + ∥ ut ∥ 2

(3.5)

(2.10) + ϵ × (3.5) yields d dt

H2 (ϕu ) + K2 (ϕu ) = 0,

t > 0,

(3.6)

where ϕu = (u, ut ) and H2 (ϕu ) =

≥ K2 (ϕu ) =

≥ ≥

1

 1 1 ∥A− 4 ut ∥2 + ∥ut ∥2 + ∥A 4 u∥2 + ∥u∥2 2     1   1  1 1 1 + ϵ A− 4 u, A− 4 ut + (u, ut ) + ∥u∥2 + η∥A 8 u∥2 + G(u)dx − A− 2 f , u Ω    2 1 θ 1 2 2 ∥ ut ∥ + 1 − √ ∥ A 4 u∥ − C 1 , 4 λ1  1  1 1 1 (1 − ϵ)∥ut ∥2 + 2η∥A 8 ut ∥2 + ϵ ∥A 4 u∥2 + ∥u∥2 + (g (u), u) − (A− 2 f , u) − ∥A− 4 ut ∥2    1  ϵ  θ  1 2 1 1−ϵ− √ ∥ut ∥2 + 2η∥A 8 ut ∥2 + ϵ 1 − √ ∥A 4 u∥ + ϵ∥u∥2 − ϵ A− 2 f , u λ1 λ1  1  2 2 4 κϵ ∥A u∥ + ∥ut ∥ − C1 ϵ

(3.7)

for given ϵ(> 0) suitably small, where and in the context C1 = C (∥f ∥V−3 ), κ > 0 denotes a small positive constant. Inserting (3.7) into (3.6), we get d dt

H2 (ϕu ) + κϵ∥ϕu ∥2E1 ≤ C1 ϵ,

t > 0.

(3.8)

Z. Yang et al. / Nonlinear Analysis 115 (2015) 103–116

111

Obviously, H2 (ϕu (0)) ≤ C (∥ϕu (0)∥2E1 + ∥f ∥2V−3 ) ≤ K .

sup H2 (ϕu (t )) ≥ −C1 ,

t ∈R+

Applying Lemma 3.3 (taking δ = κϵ, γ = ϵ there) to (3.8), we arrive at



H2 (ϕu ) ≤ sup H2 (ζ ) : ∥ζ ∥ ζ ∈E1

2 E1

≤

C1 + 1



κ

≡ K1 ,

t ≥ t0 ≡

C1 + K

ϵ

.

Therefore,

∥ϕu (t )∥2E1 ≤ K1 + C1 ,

t ≥ t0 .

(3.9)

Obviously, the estimate (3.9) holds true for the Galerkin approximation ϕun of ϕu . By the lower semi-continuity of the weak limit we know that (3.9) is really valid for the solution ϕu of problem (2.5), (2.2). By the same reason, (3.9) is valid for the limit solution of problem (2.1)–(2.2). The estimate (3.9) implies that the ball

BR = {ζ ∈ E1 |∥ζ ∥E1 ≤ R} is an absorbing set of the semigroup T (t ) for R suitably large. And BR is also an absorbing set of the semigroup Tη (t )(η > 0). Let

A = {ψ|ψ = (w) lim Tηn (tn )ϕn , where tn → ∞, ηn → 0+ , ϕn ∈ BR }. n→∞

Lemma 3.5. ϕ ∈ G if and only if one of the following conditions is satisfied: (i) ϕ0 ∈ A, and

ϕ(t ) = T (t )ϕ0 = (w) lim Tηn (t )Tηn (tn )ϕ0n = (w) lim Tηn (t + tn )ϕ0n , n→∞

n→∞

where ϕ0 = (w) limn→∞ Tηn (tn )ϕ , ϕ ∈ BR . (ii) ϕ0 ∈ E1 \ A, and n 0

n 0

ϕ(t ) = T (t )ϕ0 = (w) lim Tηn (t )ϕ0n . n→∞

We claim that the set A is the (E1 , Es ) attractor. Indeed, (i) For any ψ ∈ A,

∥ψ∥E1 ≤ lim inf ∥Tηn (tn )ϕn ∥E1 ≤ C (R), n→∞

i. e. A is bounded in E1 . For any sequence {ψ n } ⊂ A, let ψ n → ψ in E1 as n → ∞. By the definition of A,

ψ n = (w) lim Tηnk (tnk )ϕnk , k→∞

with tnk → ∞, ηnk → 0+ , ϕnk ∈ BR .

So for ϵ = 1/n, there exist ηnn < 1/n, tnn > n, ϕnn ∈ BR such that

|(Tηnn (tnn )ϕnn − ψ n , ξ )| ≤

1 n

for all ξ ∈ E1∗ ,

where E1∗ is the dual space of E1 , that is, Tηnn (tnn )ϕnn − ψ n ⇀ 0.

(3.10)

On account of ψ ⇀ ψ , we get n

Tηnn (tnn )ϕnn ⇀ ψ,

(3.11)

which means ψ ∈ A, i.e. A is a closed set in E1 . Let the sequence {ψ n } ⊂ A, ψ n → ψ in Es , with 0 ≤ s < 1. Since A is bounded in E1 , we can extract a subsequence, still denoted by {ψ n }, such that ψ n ⇀ ψ , which together with (3.10) implies (3.11), i.e. ψ ∈ A, A is also closed in Es . Therefore, A is a compact set in Es for E1 ↩→↩→ Es . (ii) For any ϕ(t ) ∈ T (t )A, by Lemma 3.5, there exists ϕ0 ∈ A such that

ϕ(t ) = T (t )ϕ0 = (w) lim Tηn (t )Tηn (tn )ϕ0n = (w) lim Tηn (t + tn )ϕ0n n→∞

where ϕ ∈ BR . So ϕ(t ) ∈ A, T (t )A ⊂ A. n 0

n→∞

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Z. Yang et al. / Nonlinear Analysis 115 (2015) 103–116

For any ϕ0 ∈ A,

ϕ0 = (w) lim Tηn (tn )ϕ0n = (w) lim Tηn (s)Tηn (tn − s)ϕ0n , n→∞

n→∞

where ϕ ∈ BR . For any fixed s ≥ 0, the sequence {Tηn (tn − s)ϕ0n } ⊂ BR is bounded in E1 , so one can extract a subsequence nk such that n 0

n

ϕ in E1 as k → ∞, Tηnk (tnk − s)ϕ0 k ⇀  which means  ϕ ∈ A and n

ϕ ∈ T (s)A. ϕ0 = (w) lim Tηnk (s)Tηnk (tnk − s)ϕ0 k = T (s) k→∞

Therefore, T (t )A = A, t ≥ 0. (iii) It is enough to show that distEs {T (t )BR , A} → 0

as t → ∞.

Or else, there exist ϵ > 0, a sequence {ϕn } ⊂ BR , tn → ∞ such that inf ∥T (tn )ϕn − ψ∥Es > ϵ.

(3.12)

ψ∈A

Then ϕn ∈ E1 \ A (or else, if ϕn ∈ A, then T (tn )ϕn ∈ A, which violates (3.12)). By Lemma 3.5,

ϕn (t ) = T (t )ϕn = (w) lim Tηnk (t )ϕnk , k→∞

ϕnk ⇀ ϕn as k → ∞

and Tηnk (t )ϕnk → ϕn (t )

in C ([0, T ], Es ) as k → ∞

(see (2.15)). For ϵ = 1/n, there exist ηnn : 0 < ηnn < 1/n, ϕnn ∈ Es such that sup ∥ϕn (t ) − Tηnn (t )ϕnn ∥Es <

t ∈[0,tn ]

1 n

.

Especially,

∥ϕn (tn ) − Tηnn (tn )ϕnn ∥Es → 0.

(3.13)

Since the sequence {Tηnn (tn )ϕnn } ⊂ BR is bounded in E1 , and E1 ↩→↩→ Es , we can extract a subsequence, still denoted by itself, such that Tηnn (tn )ϕnn →  ϕ weakly in E1 and strongly in Es ,

(3.14)

which means  ϕ ∈ A. Combining (3.13) with (3.14), we get T (tn )ϕn = ϕn (tn ) →  ϕ in Es , which violates (3.12). Theorem 3.4 is proved. 4. Global and exponential attractor in non-supercritical case

˜ 2 for the clamped boundary Theorem 4.1. Let the assumptions of Theorem 2.1 be in force (where λ1 is the first eigenvalue of Λ condition), with 1 ≤ p ≤ p˜ = (N −N2)+ . Then problem (2.1)–(2.2) (problem (2.3)–(2.4)) admits a unique weak solution u, with

ϕu = (u, ut ) ∈ Cb (R+ , E1 ) ≡ L∞ (R+ , E1 ) ∩ C (R+ , E1 ), and the solution is Lipschitz continuous in E1 , that is, ∥(z (t ), zt (t ))∥2E1 ≤ Cekt ∥(z (0), zt (0))∥2E1 ,

t ≥ 0,

(4.1)

for some C , k > 0, where z = u − v, u and v are respectively the weak solutions of Eq. (2.1) (Eq. (2.3)) corresponding to initial data (u0 , u1 ) and (v0 , v1 ). Proof. The existence of the weak solutions has been proved in Theorem 2.1 (with η = 0). So we only prove (4.1) here. Taking H-inner product by zt in (2.17), with η = 0, we have 1 d  2 dt

 1 1 ∥A− 4 zt ∥2 + ∥zt ∥2 + ∥z ∥2 + ∥A 4 z ∥2 + ∥zt ∥2 = −(g (u) − g (v), zt )

≤ C (1 + ∥u∥p2p−1 + ∥v∥p2p−1 )∥z ∥2p ∥zt ∥ ≤

1 2

1

∥zt ∥2 + C ∥A 4 z ∥2 .

Applying the Gronwall inequality to (4.2) we obtain (4.1).

(4.2)

Z. Yang et al. / Nonlinear Analysis 115 (2015) 103–116

113

Remark 4.1. (i) When 1 ≤ p ≤ p˜ , the subclass G of the limit solutions is really the set of all the weak solutions of problem (2.1)–(2.2) (problem (2.3)–(2.4)) because of the uniqueness of the weak solutions, and by (4.1), the operator semigroup T (t ) defined in (3.1) becomes the traditional continuous semigroup on E1 , that is, T (t ) : E1 → E1 ,

T (t )ϕ0 = ϕu (t ) = (u(t ), ut (t )),

where ϕu ∈ Cb (R , E1 ) as shown in Theorem 4.1. (ii) It follows from Theorem 3.4 and Remark 4.1 that the dynamical system (T (t ), E1 ) is dissipative, that is, it has a bounded absorbing set BR . Without loss of generality we assume that BR is positive invariant, that is, T (t )BR ⊂ BR for t ≥ 0. +

Theorem 4.2. Let the assumptions of Theorem 4.1 be in force, especially when p = p˜ , g ∈ C 2 (R),

|g ′′ (s)| ≤ C (1 + |s|p−2 ),

s ∈ R with p ≥ 2.

Then the following conclusions are valid: (i) The solution semigroup T (t ) possesses in E1 a compact global attractor A, which has finite fractal dimension. (ii) Any full trajectory ν = {ϕu (t ) = (u(t ), ut (t ))|t ∈ R} ⊂ A possesses the property

(u, ut , utt ) ∈ L∞ (R; V2 × V1 × H ), and there exists constant R > 0 such that sup sup(∥u(t )∥2V2 + ∥ut (t )∥2V1 + ∥utt (t )∥2 ) ≤ R2 .

ν⊂A t ∈R

(iii) The global attractor A consists of full trajectory ν = {ϕu (t )|t ∈ R} such that lim distE1 {ϕu (t ), N } = 0,

lim distE1 {ϕu (t ), N } = 0

t →−∞

t →+∞

where N is the set of all fixed points of T (t ), that is,





1



1

N = (u, 0) ∈ E1 | I + A 2 u + g (u) = A− 2 f



˜ 2 for the clamped boundary condition). Furthermore, for any ζ ∈ E1 , (replacing A by Λ lim distE1 {T (t )ζ , N } = 0.

t →+∞

(iv) The semigroup T (t ) has in E1 an exponential attractor. Lemma 4.3. Let y : R+ → R+ be an absolutely continuous function satisfying d dt

y(t ) + 2ϵ y(t ) ≤ h(t )y(t ) + z (t ),

where ϵ > 0, z ∈ L1loc (R+ ), y(t ) ≤ e

m



y(0)e

−ϵ t

t s

t > 0,

h(τ )dτ ≤ ϵ(t − s) + m for t ≥ s ≥ 0 and some m > 0. Then t



−ϵ(t −τ )

|z (τ )|e

+

dτ



,

t > 0.

0

Lemma 4.4 (Quasi-stability). Let the assumptions of Theorem 4.1 be valid, u, v be the solutions of problem (2.1)–(2.2) (problem (2.3)–(2.4)) with initial data in BR . Then z = u − v satisfies the relation

∥(z (t ), zt (t ))∥2E1 ≤ C ∥(z (0), zt (0))∥2E1 e−κ t + K sup ∥z (s)∥2

(4.3)

0≤s≤t

for some constants C , K > 0. Proof. (i) When 1 ≤ p < p˜ , taking H-inner product by zt + ϵ z in (2.17), with η = 0, we get d dt

1



1

H4 (z , zt ) + (1 − ϵ)∥zt ∥2 − ϵ∥A− 4 zt ∥2 + ϵ ∥z ∥2 + ∥A 4 z ∥2



= −(g (u) − g (v), zt + ϵ z ),

where H4 (z , zt ) =

1 2

1

1

∥A− 4 zt ∥2 + ∥zt ∥2 + (1 + ϵ)∥z ∥2 + ∥A 4 z ∥2 + 2ϵ

∼ ∥(z , zt )∥2E1



1



A− 2 + I zt , z



(4.4)

114

Z. Yang et al. / Nonlinear Analysis 115 (2015) 103–116

for ϵ > 0 suitably small. On account of p < p˜ , V1−δ ↩→ L2p for δ : 0 < δ ≪ 1 and the interpolation theorem we have the control p−1 |(g (u) − g (v), zt + ϵ z )| ≤ C (1 + ∥u∥p2p−1 + ∥v∥2p )(∥z ∥2p ∥zt ∥ + ϵ∥z ∥2p ∥z ∥)

≤ C ∥z ∥V1−δ (∥zt ∥ + ϵ∥z ∥) 1 ϵ 1 ≤ ∥zt ∥2 + ∥A 4 z ∥2 + C ∥z ∥2 .

2 2 Therefore, there exists constant κ > 0 such that d dt

H4 (z , zt ) + κ H4 (z , zt ) ≤ C ∥z ∥2 ,

∥(z (t ), zt (t ))∥2E1 ≤ C ∥(z (0), zt (0))∥2E1 e−κ t + C

t



e−κ(t −τ ) ∥z (τ )∥2 dτ

0

≤ C ∥(z (0), zt (0))∥2E1 e−κ t + K sup ∥z (τ )∥2 ,

(4.5)

0≤τ ≤t

where K = C /κ . (ii) When p = p˜ , rewriting (4.4) in the form d dt

H4 (z , zt ) + 1

=

2

1 2

   1 1 (g (u) − g (v), z ) + (1 − ϵ)∥zt ∥2 − ϵ∥A− 4 zt ∥2 + ϵ ∥z ∥2 + ∥A 4 z ∥2 + ϵ(g (u) − g (v), z )

(˜g (u, v), z 2 ),

(4.6)

where g˜ (u, v) =

1



g ′′ (λu + (1 − λ)v)(λut + (1 − λ)vt )dλ. 0

Since V1 ↩→ L2p , there exists constant l > 0 such that

|(g (u) − g (v), z )| ≤ C (1 + ∥u∥p2p−1 + ∥v∥p2p−1 )∥z ∥2p ∥z ∥ ≤ C ∥z ∥V1 ∥z ∥ ≤

1 2

∥z ∥2V1 + l∥z ∥2 ,

which means 1

(g (u) − g (v), z ) + l∥z ∥2 ≥ − ∥z ∥2V1 , 2

and

|(˜g (u, v), z 2 )| ≤ C (1 + ∥u∥p2p−2 + ∥v∥p2p−2 )(∥ut ∥ + ∥vt ∥)∥z ∥22p ≤ C (∥ut ∥ + ∥vt ∥)∥z ∥2V1 we infer from (4.6) that d dt

H5 (z , zt ) +

1 2

∥zt ∥2 + K5 (z , zt ) ≤ C (∥ut ∥ + ∥vt ∥)∥z ∥2V1 + l(z , zt ) 1

≤ C (∥ut ∥ + ∥vt ∥)H5 (z , zt ) + ∥zt ∥2 + l2 ∥z ∥2 , 2

(4.7)

where 1

 1 (g (u) − g (v), z ) + l∥z ∥2 ∼ ∥zt ∥2 + ∥A 4 z ∥2 , 2     ϵ 1 1 K5 (z , zt ) = −ϵ− √ ∥zt ∥2 + ϵ ∥z ∥2 + ∥A 4 z ∥2 + (g (u) − g (v), z ) 2 λ1 ≥ 2κ H5 (z , zt ) − lϵ∥z ∥2

H5 (z , zt ) = H4 (z , zt ) +

(4.8)

for ϵ > 0 suitably small. Inserting (4.8) into (4.7), we get d dt

H5 (z , zt ) + 2κ H5 (z , zt ) ≤ C (∥ut ∥ + ∥vt ∥)H5 (z , zt ) + l2 ∥z ∥2 .

(4.9)

By (2.11), with η = 0, there exists m > 0 such that t



(∥ut (τ )∥ + ∥vt (τ )∥)dτ ≤ C

C s

t



(∥ut (τ )∥2 + ∥vt (τ )∥2 )dτ s

≤ κ(t − s) + m for t ≥ s ≥ 0.

1/2

(t − s)1/2

Z. Yang et al. / Nonlinear Analysis 115 (2015) 103–116

115

Applying Lemma 4.3 to (4.9), we get

∥(z (t ), zt (t ))∥

2 E1

m



≤ Ce

∥(z (0), zt (0))∥

2 −κ t E1 e

+l

2

t



e

−κ(t −τ )

∥z (τ )∥ dτ 2



0

≤ Cem ∥(z (0), zt (0))∥2E1 e−κ t + K sup ∥z (τ )∥2 , 0≤τ ≤t

where K = Ce l /κ . Lemma 4.4 is proved. m 2

Proof of Theorem 4.2. The estimates (4.1) and (4.3) show that the dissipative system (T (t ), E1 ) is quasi-stable on the absorbing set BR , so the conclusions (i) and (ii) follow directly from the standard theory on global attractor (cf. Theorems 7.9.4–7.9.6 and 7.9.8 in [13]). The energy equality (2.10) (with η = 0) shows that H (u, ut ) is a strictly Lyapunov function on E1 , so the dynamical system (T (t ), E1 ) is gradient, and by conclusion (ii), it has a compact global attractor. Therefore, the conclusion (iii) of Theorem 2.2 holds (cf. Theorems 2.28 and 2.31 in [12]). We see from the conclusion (ii) that the global attractor A is included and bounded in E2 = V2 × V1 . Let D be the closure of the 1-neighborhood of A in E2 , that is,

D = [{ζ ∈ E2 |distE2 {ζ , A} ≤ 1}]E1 . Then D is bounded in E2 and closed in E1 , and it is an absorbing set of T (t ), without loss of generality we assume that T (t )D ⊂ D , t ≥ 0. By Lemma 4.4, T (t ) is quasi-stable on D . For every ϕ0 ∈ D , ϕ(t ) = S (t )ϕ0 = (u(t ), ut (t )) ∈ D and by Eq. (2.1), ∥utt ∥ ≤ C (D ),

∥T (t2 )ϕ0 − T (t1 )ϕ0 ∥E1 ≤



t2

∥ϕ ′ (t )∥E1 dt ≤ C (D )|t2 − t1 |.

t1

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