Optimal attractors of the Kirchhoff wave model with structural nonlinear damping

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Optimal attractors of the Kirchhoff wave model with structural nonlinear damping ✩ Yanan Li, Zhijian Yang ∗ School of Mathematics and Statistics, Zhengzhou University, No. 100, Science Road, Zhengzhou 450001, China Received 21 January 2019; revised 26 October 2019; accepted 20 November 2019

Abstract The paper investigates the well-posedness and longtime dynamics of the Kirchhoff wave model with structural nonlinear damping: utt − φ(∇u2 )u + σ (∇u2 )(−)θ ut + f (u) = g(x), with θ ∈ [1/2, 1). N+2θ We find a new critical exponent p ∗ ≡ N+2 N−2 (> pθ ≡ N−2 , N ≥ 3) and show that when the growth exponent p of the nonlinearity f (u) is up to the range: 1 ≤ p < p∗ : (i) the IBVP of the equation is well-posed and its solution is of additionally global regularity when t > 0; (ii) for each θ ∈ [1/2, 1), the related solution semigroup has in natural energy space H an optimal global attractor Aθ whose compactness and attractiveness are in the regularized space H1+θ where Aθ lies, and an optimal exponential attractor Eθ∗ whose compactness, boundedness of the fractional dimension and the exponential attractiveness are in H1+θ where Eθ∗ lies, respectively; (iii) the family of global attractors {Aθ }θ∈[1/2,1) is upper semi-continuous at each point θ0 ∈ [1/2, 1). The paper breaks though the longstanding existed growth restriction: 1 ≤ p ≤ pθ for pθ had been considered a uniqueness index, deepens and extends the results in literature [6,25,27]. © 2019 Elsevier Inc. All rights reserved.

MSC: 35B33; 35B41; 35B65; 37L15; 37L30 Keywords: Kirchhoff wave model; Structural nonlinear damping; Well-posedness; Optimal global attractor; Optimal exponential attractor; Upper semicontinuity

✩

Supported by National Natural Science Foundation of China (No. 11671367).

* Corresponding author.

E-mail addresses: [email protected] (Y. Li), [email protected] (Z. Yang). https://doi.org/10.1016/j.jde.2019.11.084 0022-0396/© 2019 Elsevier Inc. All rights reserved.

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1. Introduction In this paper, we investigate the well-posedness and longtime dynamics of the Kirchhoff wave model with structural nonlinear damping utt − φ(∇u2 )u + σ (∇u2 )(−)θ ut + f (u) = g(x) in  × R+ ,

(1.1)

u|∂ = 0, u(x, 0) = u0 (x), ut (x, 0) = u1 (x), x ∈ ,

(1.2)

where θ ∈ [1/2, 1) is said to be a dissipative index,  is a bounded domain in RN with the smooth boundary ∂,  ·  stands for the norm in L2 (), φ(s), σ (s) and f (s) are nonlinear functions specified later, and g(x) is an external force term. When N = 1, φ(s) = a + bs, σ (s) ≡ 0, f (s) ≡ 0, the corresponding Eq. (1.1) was established by Kirchhoff [14] to model small vibrations of an elastic stretched string. After that, there have been extensive researches on the well-posedness and asymptotic behavior of Kirchhoff wave models with different types of linear dissipation (cf. [1,2,12,19–21,24] and references therein). Global and exponential attractors are two basic concepts of studying longtime dynamics of dissipative dynamical system. When θ ∈ [0, 1], σ ≡ σ0 > 0 (linear damping case), the attractors of the corresponding problem (1.1)-(1.2) had been studied by some authors under the assumptions that the source term f (u) is either absent or subcritical (cf. [18,19,25] and references therein). Pioneering researches for model (1.1) originate from [6,7,15,17]. Medeiros and Miranda [17] and then Lazo [15] proved the existence of weak solutions of problem (1.1)-(1.2) by taking 0 < θ ≤ 1 and f (s) ≡ 0. However, the uniqueness was only shown when 1/2 ≤ θ ≤ 1. Chueshov [6] worked with the same restriction on θ to have studied the well-posedness and longtime dynamics for an abstract second order evolution equation of the form utt + φ(A1/2 u2 )Au + σ (A1/2 u2 )Aθ ut + F (u) = 0, t > 0,

(1.3)

where A is an operator defined in a Hilbert space H . The main motivating example of Eq. (1.3) is Eq. (1.1). By the way, the uniqueness of weak solutions of model (1.1) with 0 < θ < 1/2 is still an open problem. When θ = 1, Chueshov [7] studied the Kirchhoff wave model with strong nonlinear damping utt − φ(∇u2 )u − σ (∇u2 )ut + f (u) = g(x).

(1.4)

N+4 + = A major breakthrough is that he found a supercritical exponent p∗∗ ≡ (N−4) + , where a max{a, 0}, and showed that when the growth exponent p of the source term f (u) is up to the supercritical range: p ∗ < p < p ∗∗ , problem (1.4)-(1.2) is still well-posed and the related solution semigroup has a finite-dimensional global attractor in phase space H = H01 ∩ Lp+1 × L2 in the sense of “partially strong topology”, where p∗ ≡ N+2 N−2 is said to be a critical exponent for 1 p+1 ∗ as p ≤ p but the embedding fails as p > p ∗ . Recently, Ding, Yang and Li [10] H0 → L further removed the restriction of “partially strong topology” and improved the results in [7]. Ma and Zhong [16] showed that when 1 ≤ p ≤ p∗ , the solution semigroup associated with problem (1.4)-(1.2) has in phase space H = H01 × L2 an optimal global attractor A whose compactness and attractiveness are in the regularized space H01 × H01 where A lies.

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When 1/2 ≤ θ < 1, Chueshov [6] also found an exponent pθ ≡ N+2θ N−2 depending on the dissipative index θ and proved the well-posedness and the existence of global and exponential attractors for the Kirchhoff wave model (1.1) provided that 1 ≤ p ≤ pθ . Obviously, the results obtained in [6] are weaker than those in [7] for the strength of the damping (dissipation) becomes weaker. For the related researches on this type of models, one can see [8,22]. Comparing the results in [6] with those in [7], one easily finds that when θ ∈ [1/2, 1), maybe the restrictive exponent pθ is not optimal. Recently, for the structural linear damping case, i.e., N+4θ σ (s) ≡ 1, Yang, Ding and Li [25] have found an optimal supercritical exponent pθ ≡ (N−4θ) + and showed the well-posedness, the existence and regularity of the global and exponential attractors provided that 1 ≤ p ≤ pθ and θ ∈ (1/2, 1). We guess that even for the structural nonlinear damping case, the exponent pθ is also not optimal relative to phase space H = H01 × L2 , the optimal critical exponent of ensuring the well-posedness, the existence and regularity of the global and exponential attractors should be p ∗ (> pθ ) rather than pθ . The stability of perturbed attractors has been extensively concerned in the last years (cf. [5, 9,11] and references therein). Recently, the upper semicontinuity of attractors on the dissipative index θ has been studied for some evolution models (cf. [4,26,27] and references therein). Yang and Li [27] have proved the upper semicontinuity of the global attractors obtained in [25] on the dissipative index θ ∈ (1/2, 1). Unfortunately, the arguments in [27] does not hold for θ = 1/2, that is, whether this upper semicontinuity holds at θ0 = 1/2 is still unsolved. In this paper, we continue to investigate this topic. The purpose of the present paper is to verify above-mentioned conjecture. We show that when the growth exponent p of the nonlinearity f (u) is up to the range: 1 ≤ p < p ∗ ≡ N+2 N−2 (> pθ , N ≥ 3): (i) problem (1.1)-(1.2) is well-posed and its solution is of additionally global regularity when t > 0; (see Theorem 2.6) (ii) for each θ ∈ [1/2, 1), the related solution semigroup has in natural energy space H an optimal global attractor Aθ whose compactness and attractiveness are in the regularized space H1+θ where Aθ lies, and an optimal exponential attractor Eθ∗ whose compactness, boundedness of the fractional dimension and the exponential attractiveness are in H1+θ where Eθ∗ lies, respectively; (see Theorem 5.3) (iii) the family of global attractors {Aθ }θ∈[1/2,1) is upper semi-continuous on the dissipative index θ at each point θ0 ∈ [1/2, 1). (see Theorem 3.3) In the present paper, by using more delicate estimate technique than before we break though the longstanding existed growth restriction: 1 ≤ p ≤ pθ = N+2θ N−2 (N ≥ 3) for pθ had been considered a uniqueness index, deepen and extend the results in literature [6]. We restrict ourselves to the case N ≥ 3 because pθ = +∞ when N = 1, 2, and in this case the well-posedness and the existence of global and exponential attractors had been established in [6]. However, the other results in the present paper, for example: the upper semicontinuity of the perturbed attractor Aθ on the dissipative index θ , and the existence of optimal global and exponential attractors, still hold for the cases N = 1, 2. We mention that the results in the present paper hold for the case σ (s) ≡ 1, so the results, such as the upper semicontinuity of global attractor Aθ on θ at θ0 = 1/2, and the existence of optimal global and exponential attractors, also deepen and extend those in [25,27]. This paper is arranged as follows. In Section 2, we discuss the well-posedness and the additional regularity of weak solutions. In Section 3, we investigate the existence of global attractors

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Aθ and their upper semicontinuity on dissipative index θ . In Section 4, we show the existence of exponential attractor. In Section 5, we study the optimal global and exponential attractors. 2. Well-posedness We begin with the following abbreviations: Lp = Lp (), H k = H k (), V1 = H01 (), V−1 = H −1 ,  ·  =  · L2 , with p ≥ 1, H k are the L2 -based Sobolev spaces. The notation (·, ·) for the L2 -inner product will also be used for the notation of duality pairing between dual spaces, the sign X1 → X2 denotes that the space X1 continuously embeds into X2 and X1 →→ X2 denotes that X1 compactly embeds into X2 , and C(· · · ) stands for positive constants depending on the quantities appearing in the parenthesis. Obviously, V1 →→ L2 →→ V−1 . Define the operator A : V1 → V−1 , (Au, v) = (∇u, ∇v), ∀u, v ∈ V1 . Then A is self-adjoint in L2 and strictly positive on V1 , and we can define the powers As of A s for all s ∈ R and the Hilbert spaces Vs = D(A 2 ) with the inner products and the norms: s

s

s

(u, v)s = (A 2 u, A 2 v), us = uVs = A 2 u, s ∈ R, respectively. Rewriting Eq.(1.1) at an abstract level, we get utt + φ(u21 )Au + σ (u21 )Aθ ut + f (u) = g,

(2.1)

u(0) = u0 , ut (0) = u1 .

(2.2)

Define the phase spaces H = V1 × L2 , H1+θ = V1+θ × Vθ , Hθ = Vθ × V−θ , with θ ∈ [1/2, 1), which are equipped with the usual graph norms, for example, (u, v)2H = u21 + v2 , (u, v) ∈ H. Obviously, H1+θ →→ H →→ Hθ . Assumption 2.1. (1) σ, φ ∈ C 1 (R+ ), σ (s) > 0, φ(s) > 0, ∀s ∈ R+ , and μφ := lim inf φ(s) > 0. s→+∞

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(2) f ∈ C 1 (R) satisfying f (0) = 0, |f  (s)| ≤ C(1 + |s|p−1 ) for some 1 ≤ p < p ∗ :=

N +2 , N −2

(2.3)

and μf := lim inf |s|→∞

f (s) > −λ1 μφ , s

(2.4)

where λ1 (> 0) is the first eigenvalue of the operator A. (3) g ∈ L2 , (u0 , u1 ) ∈ H with (u0 , u1 )H ≤ R. Remark 2.1. Assumption 2.1: (1)-(2) imply that there exists a 0 < δ0 < min{

μφ 3 , 1}

such that

λ1 μφ − 3λ1 δ0 2 s − C(δ0 ), f (s)s ≥ −(λ1 μφ − 3λ1 δ0 )s 2 − C(δ0 ), s ∈ R, 2 (s) ≥ (μφ − δ0 )s − C(δ0 ), φ(s)s ≥ (μφ − δ0 )s − C(δ0 ), s ∈ R+ , F (s) ≥ −

where F (s) =

s 0

f (r)dr and (s) =

s 0

φ(r)dr.

Lemma 2.2. [23] Let X, B and Y be three 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). Lemma 2.3. [3] Let X be a Banach space, the set Z ⊂ C(R+ ; X), and the mapping : X → R+ satisfy (v(0)) ≤ C for some C ≥ 0 and every v ∈ Z. In addition, let for every v ∈ Z the function t → (v(t)) be continuously differentiable, and satisfy the differential inequality d (v(t)) + κv(t)2X ≤ ω dt for some ω > 0 and κ > 0 independent v ∈ Z. Then, for every δ > 0 there exists a tδ > 0 such that   (v(t)) ≤ sup (v) : κv2X ≤ ω + δ , ∀t ≥ tδ . v∈Z

Lemma 2.4. [25] Let {xn } be a bounded sequence and ψ ∈ C(R) be a monotone function. Then ψ(lim inf xn ) ≤ lim inf ψ(xn ). n→∞

n→∞

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Definition 2.5. A function u is said to be a weak solution of problem (2.1)-(2.2) on an interval [0, T ] if u ∈ L∞ (0, T ; V1 ), ut ∈ L∞ (0, T ; L2 ) ∩ L2 (0, T ; Vθ ), and u satisfies Eq. (2.1) in the sense of distribution. Theorem 2.6. (Well-posedness) Let Assumption 2.1 be valid. Then for every T > 0 and θ ∈ [1/2, 1), problem (2.1)-(2.2) admits a unique weak solution uθ , with (uθ , uθt ) ∈ C([0, T ]; H) ∩ L2 (0, T ; V2−θ × Vθ ), uθtt ∈ L∞ (0, T ; V−2θ ), and (u

θ

(t), uθt (t))2H

+ uθtt (t)2−2θ

t+1 t θ 2 + u (s)2−θ ds + uθt (s)2θ ds ≤ C(R), t ≥ 0. (2.5) t

0

Moreover, this solution possesses the following properties: (i) (Dissipativity) There exists a positive constant R0 independent of θ ∈ [1/2, 1) such that (uθ (t), uθt (t))H ≤ R0 , ∀t ≥ t (R),

(2.6)

where t (R) > 0 is a constant depending only on R. (ii) (Energy identity) The following energy identity t θ

E(u

(t), uθt (t)) +

σ (uθ (τ )21 )uθt (τ )2θ dτ = E(uθ (s), uθt (s))

(2.7)

s

holds for every t ≥ s ≥ 0, where  1 θ 2 ut  + (uθ 21 ) + (F (uθ ), 1) − (g, uθ ). 2

E(uθ , uθt ) =

(iii) (Lipschitz stability in weaker space) Let uθ,1 and uθ,2 be two weak solutions of problem (2.1)-(2.2), zθ = uθ,1 − uθ,2 . Then t (z

θ

(t), ztθ (t))2Hθ

+



 zθ (s)21 + ztθ (s)2 ds ≤ C(R)eC(R)t (zθ (0), ztθ (0))2Hθ ,

0

∀t ≥ 0. (iv) (Global regularity when t > 0) For any 0 < a < T , (uθ , uθt , uθtt ) ∈ L∞ (a, T ; V1+θ × Vθ × V−θ ) ∩ L2 (a, T ; V2 × V1 × L2 ), and

(2.8)

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u

θ

(t)21+θ

+ uθt (t)2θ

+ uθtt (t)2−θ

+

t+1 t

7

uθ (s)22 + uθt (s)21 + uθtt (s)2 ds (2.9)



 −1  ≤ C(R) 1 + t 1−θ 1 + q(θ) + t eC(R)t , ∀t > 0,

where q(θ) is a nonnegative continuous function on [1/2, 1) with limθ→1 q(θ) = +∞. Proof. For simplicity, we omit the superscript θ of uθ in the following. We first give some a priori estimates to the solutions of problem (2.1)-(2.2). Using the multiplier ut in Eq. (2.1) and integrating the resulting expression over (0, t) yield t E(u(t), ut (t)) +

σ (u(τ )21 )ut (τ )2θ dτ = E(u0 , u1 ) ≤ C(R), t ≥ 0.

(2.10)

0

By Remark 2.1, 1 (u21 ) + 2

 F (u)dx≥ 

 λ1 μφ − 3λ1 δ0 1

u21 − C(δ0 ) (μφ − δ0 )u21 − 2 λ1

= δ0 u21 − C(δ0 ), so C0 (u(t), ut (t))2H − C ≤ E(u(t), ut (t)) ≤ C(R), t ≥ 0, where C0 = (2.10)-(2.11),

δ0 2.

(2.11)

Since σ (s) > 0 and (2.11), σ (u(t)21 ) ≥ σR > 0, ∀t ≥ 0. By estimates t

t ut (τ )2θ dτ

σR

≤

0

σ (u(τ )21 )ut (τ )2θ dτ ≤ C(R), t ≥ 0.

(2.12)

0

Taking account of the embedding L

1+ p1

→ V−1 → V−2θ , we infer from Eq. (2.1) that

 utt 2−2θ ≤ C(R) u22−2θ + ut 2 + f (u)2 1+ 1 + g2 L p  2p+2 ≤ C(R) u21 + u1 + ut 2 + g2 + 1 ≤ C(R), t ≥ 0.

(2.13)

Using the multiplier A1−θ u in Eq. (2.1) yields d (ut , A1−θ u) + σ (u21 )(Aθ ut , A1−θ u) + φ(u21 )u22−θ + (f (u), A1−θ u) dt =ut 21−θ

+ (g, A

1−θ

u).

(2.14)

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Due to φ(s) > 0 and estimate (2.11), we obtain φ(u(t)21 ) ≥ φ0 > 0, ∀t ≥ 0, and σ (u2 )(Aθ ut , A1−θ u) ≤ C(R)ut θ u2−θ ≤ φ0 u2 + C(R)ut 2 , θ 1 2−θ 4 1−θ (ut , A u) ≤ ut u2−2θ ≤ Cut u1 ≤ C(R), (g, A1−θ u) ≤ Cgu2−2θ ≤ φ0 u2 + C, 2−θ 4 

 p |(f (u), A1−θ u)| ≤ C |u| + |u|p |A1−θ u|dx ≤ Cu21 + CA1−θ u 2N u L N−2θ



≤ C(R) + Cu2−θ u

p 2Np L N+2θ

≤

φ0 u22−θ + C(R), ∀t ≥ 0, 4

where we have used the fact: ⎧ p p Cu 2N ≤ Cu1 , ⎪ ⎪ ⎨ N−2 L p u 2Np ≤ ⎪ pα p(1−α) pα ⎪ L N+2θ ⎩ up(1−α) u ≤ Cu1 u2−θ , 2N 2N L N−2

≤ and where pα =

L N−2(2−θ)

2Np

L N+2θ

if 1 ≤ p ≤ if

N+2θ N−2

N+2θ N−2 ,

< p < p∗ ,

(2.15)

φ0 u2−θ + C(R), 8C

p(N−2)−(N+2θ) 2(1−θ)

< 1. Inserting above estimates into (2.14) gives



d φ0 (ut , A1−θ u) + u22−θ ≤ C(R) 1 + ut 2θ , dt 4 t+1 t+1

 φ0 2 1 + ut (s)2θ ds ≤ C(R), t ≥ 0, u(s)2−θ ds ≤ C(R) + C(R) 4 t

(2.16)

t

where we have used estimate (2.12). The combination of (2.11)-(2.13) and (2.16) gives estimate (2.5). Obviously, estimate (2.5) holds for the Galerkin approximations un , so there exists an element u, with (u, ut ) in L∞ (0, T ; H) ∩ L2 (0, T ; V2−θ × Vθ ) such that (subsequence if necessary) (un , unt ) → (u, ut ) weakly∗ in L∞ (0, T ; H) ∩ L2 (0, T ; V2−θ × Vθ ). And by Lemma 2.2, (un , unt ) → (u, ut ) in C([0, T ]; V1−δ × V−δ ), with δ : 0 < δ  1; un → u in L2 (0, T ; V1 ) and a.e. (x, t) ∈ QT =  × [0, T ]; f (un ) → f (u) weakly in L unt

→ ut in L (QT ). 2

1+ p1

(QT );

(2.17)

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Consequently, T

φ(un (t)2 ) − φ(u(t)2 ) 2 dt 1

1

0

≤

T 1

 2  φ λun (t)2 + (1 − λ)u(t)2 dλ un (t)2 − u(t)2 dt 1

0

1

1

1

0

T ≤C(R)

2 un (t)1 + u(t)1 un (t) − u(t)21 dt

0

T un (t) − u(t)21 dt → 0,

≤C(R) 0

and for any ψ ∈ C0∞ (), T

 φ(un 21 )Aun − φ(u21 )Au, ψ dt

0

T



=

T



φ(un 21 ) − φ(u21 )

0



≤C(R, T )

(Au , ψ)dt +

φ(u21 )(Aun − Au, ψ)dt

n

0

φ(un 21 ) − φ(u21 )L2 (0,T )

 + un − uL2 (0,T ;V1 ) → 0.

Similarly, T

σ (un (t)2 ) − σ (u(t)2 ) 2 dt → 0, 1

1

0

and for any ψ ∈ C0∞ (), T

 σ (un 21 )Aθ unt − σ (u21 )Aθ ut , ψ dt → 0.

0

Therefore, the limiting function u is a weak solution of problem (2.1)-(2.2) and satisfies estimate (2.5). (i) (Dissipativity) Using the multiplier ut + u in Eq. (2.1) gives  d (u, ut ) + σ (u21 )ut 2θ +  φ(u21 )u21 + (f (u), u) dt  = (g, u) + ut 2 − σ (u21 )(Aθ ut , u) ,

(2.18)

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where (u, ut ) = E(u, ut ) + (ut , u) ≥

C0 (u, ut )2H − C 2

for  > 0 suitably small. Obviously, σR ≤ σ (u21 ) ≤ C(R) and (u0 , u1 ) ≤ C(R). It follows from Remark 2.1 and estimate (2.5) that

λ1 μφ − 3λ1 δ0  φ(u21 )u21 + (f (u), u)≥ μφ − δ0 − u21 − C(δ0 ) ≥ 2δ0 u21 − C, λ1 σ (u2 )(Aθ ut , u) ≤ C(R)ut θ uθ ≤ δ0 u2 + C(R)ut 2 . θ 1 1 4λ1 Inserting above estimates into (2.18) we have that there exists a constant R > 0 such that when  ∈ (0, R ), d (u, ut ) + κ(u, ut )2H ≤ K, dt where κ and K are positive constants independent of R and θ . Therefore, applying Lemma 2.3 we obtain estimate (2.6). 2N

(ii) (Energy identity and (u, ut ) ∈ C([0, T ]; H)) By the embedding L N+2θ → V−θ , (2.5) and (2.15), f (u)2−θ ≤ Cf (u)2

2N

L N+2θ

≤ C(u21 + u

2p 2Np

L N+2θ

) ≤ C(R)(1 + u22−θ ),

t+1 f (u)2−θ ds ≤ C(R), ∀t ≥ 0, t

that is, f (u) ∈ L2 (0, T ; V−θ ). So utt = −φ(u21 )Au − σ (u21 )Aθ ut − f (u) + g ∈ L2 (0, T ; V−θ ). Then, taking the multiplier ut (∈ L2 (0, T ; Vθ )) in Eq. (2.1) we obtain energy identity (2.7). For any t ∈ [0, T ], it follows from energy identity (2.7) and convergence (2.17) that lim E(u(s), ut (s)) = E(u(t), ut (t));

s→t

(u, ut ) ∈ C([0, T ]; V1−δ × V−δ ) ∩ L∞ (0, T ; H), with δ : 0 < δ  1; u(x, s) → u(x, t) a.e. x ∈  as s → t, which imply that (u, ut ) ∈ Cw ([0, T ]; H), and

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lim (g, u(s)) = (g, u(t)), (u(t), ut (t))2H ≤ lim inf (u(s), ut (s))2H ,

s→t

s→t

(u(t)21 ) ≤ (lim inf u(s)21 ) ≤ lim inf (u(s)21 ), s→t s→t

 

 λ1 μφ λ1 μφ 2 |u(t)| dx ≤ lim inf F (u(s)dx + u(t)2 , F (u(t)) + s→t 2 2





where we have used Remark 2.1, Lemma 2.4 and the Fatou lemma. Therefore, 

1 1 lim sup ut (s)2 + lim inf (u(s)21 ) + (F (u(s)), 1) s→t 2 s→t 2 

1 ≤ lim ut (s)2 + (u(s)21 ) + (F (u(s)), 1) s→t 2 1 = ut (t)2 + (u(t)21 ) + (F (u(t)), 1) 2 

1 1 ≤ lim inf ut (s)2 + lim inf (u(s)21 ) + (F (u(s)), 1) , s→t 2 s→t 2 which means ut (t)2 = lim ut (s)2 . s→t

(2.19)

Similarly, (u(t)21 ) = lim (u(s)21 ). s→t

Thus by the fact: φ(u21 ) ≥ φ0 we have 0 ≤ φ0 u(s)21 − u(t)21 ≤ (u(s)21 ) − (u(t)21 ) → 0 as s → t, that is, u(t)21 = lim u(s)21 . s→t

(2.20)

By the uniform convexity of H, (2.19)-(2.20) and (u, ut ) ∈ Cw ([0, T ]; H), we have (u, ut ) ∈ C([0, T ]; H). (iii) (Lipschitz stability in weaker space) Let z = u1 − u2 , ui be a weak solution of problem (2.1)-(2.2) with initial data (ui (0), uit (0))H ≤ R. Then z solves 1 1 ztt + φ12 Az + σ12 Aθ zt + f (u1 ) − f (u2 ) = I (u1 , u2 ), 2 2

(2.21)

(z(0), zt (0)) = (u1 (0), u1t (0)) − (u2 (0), u2t (0)),

(2.22)

where φ12 = φ1 + φ2 , σ12 = σ1 + σ2 with φi = φ(ui 21 ) and σi = σ (ui 21 ), i = 1, 2 and

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Y. Li, Z. Yang / J. Differential Equations ••• (••••) •••–•••

12

1 1 I (u1 , u2 ) = − (φ1 − φ2 )A(u1 + u2 ) − (σ1 − σ2 )Aθ (u1t + u2t ). 2 2 Taking the multiplier z + A−θ zt in Eq. (2.21) gives 4



  1 d Ij , H (z, zt ) + φ12 z21 + σ12 − 2 z2 + (φ1 − φ2 ) A(u1 + u2 ), z = 2 dt

(2.23)

j =1

where H (z, zt ) = 2zt 2−θ + φ12 z21−θ + 4(zt , z) + σ12 z2θ ,  I1 = φ  (u1 21 )(Au1 , u1t ) + φ  (u2 21 )(Au2 , u2t ) z21−θ ,  I2 =  σ  (u1 21 )(Au1 , u1t ) + σ  (u2 21 )(Au2 , u2t ) z2θ , 

 I3 = −2 I (u1 , u2 ), A−θ zt − (σ1 − σ2 ) Aθ (u1t + u2t ), z , 

I4 = −2 f (u1 ) − f (u2 ), z + A−θ zt . By (2.5) and Assumption 2.1, φi ≥ φ0 , σi ≥ σR and 2 

 φi + |φi | + σi + |σi | ≤ C(R), φ1 − φ2 + σ1 − σ2 ≤ C(R)z1 .

(2.24)

i=1

Thus, there exist positive constants Ci , i = 1, 2 such that C1 (z, zt )2Hθ ≤ H (z, zt ) ≤ C2 (z, zt )2Hθ for  > 0 suitably small. Let β(t) =

2  

 ui (t)22−θ + uit (t)2θ .

i=1

By estimates (2.5), (2.24)-(2.25),

 (φ1 − φ2 ) A(u1 + u2 ), z 1   2 ≤ φ  λu1 21 + (1 − λ)u2 21 dλ A(u1 + u2 ), z 0

≤C(R, )u1 + u2 22−θ z2θ ≤ C(R, )β(t)z2θ , and

(2.25)

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Y. Li, Z. Yang / J. Differential Equations ••• (••••) •••–•••

13

 |I1 | ≤ C(R) u1 2−θ u1t θ + u2 2−θ u2t θ z21−θ ≤ C(R)β(t)z2θ ,  |I2 | ≤ C(R) u1 2−θ u1t θ + u2 2−θ u2t θ z2θ ≤ C(R)β(t)z2θ ,

  |I3 | ≤ C(R)z1 u1 2−θ + u1t θ + u2 2−θ + u2t θ zθ + zt −θ φ0 z21 + C(R, )β(t)H (z, zt ), 2 

  |I4 | ≤ C 1 + |u1 |p−1 + |u2 |p−1 |z| |z| + |A−θ zt | dx ≤





≤C

  1 + |u1 |p−1 + |u2 |p−1 |z|2 + |A−θ zt |2 dx





  p−1 p−1 ≤ C 1 + u1 Lp+1 + u2 Lp+1 z2Lp+1 + A−θ zt 2Lp+1   ≤ C(R) z21−δ + zt 21−δ−2θ φ 0 z21 + zt 2 + C(R, )H (z, zt ), ≤ 2 where we have used the Sobolev embedding: V1−δ → Lp+1 with δ : 0 < δ  1 for 1 ≤ p < p ∗ . Inserting above estimates into (2.23) yields 

  d H (z, zt ) + κ z21 + zt 2 ≤ C(R) 1 + β(t) H (z, zt ) dt

(2.26)

for  > 0 suitably small, where (and in the following) κ denotes a small positive constant. Applying the Gronwall lemma to (2.26) gives desired estimate (2.8). (iv) (Global regularity when t > 0) Taking the multiplier Au + A1−θ ut in Eq. (2.1) turns out 3

  1 d πj , H1 (u, ut ) + φ0 u22 + (σR − )ut 21 +  f (u), Au ≤ 2 dt

(2.27)

j =1

where we have used the fact: φ(u21 ) ≥ φ0 and σ (u21 ) ≥ σR , and where H1 (u, ut ) = ut 21−θ + φ(u21 )u22−θ + σ (u21 )u21+θ + 2(ut , Au),  π1 = σ  (u21 )u21+θ + φ  (u21 )u22−θ (Au, ut ), 

π2 = g, Au + A1−θ ut , 

π3 = − f (u), A1−θ ut . By estimate (2.5),   aR u21+θ + ut 21−θ ≤ H1 (u, ut ) ≤ bR u21+θ + ut 21−θ

(2.28)

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Y. Li, Z. Yang / J. Differential Equations ••• (••••) •••–•••

14

for  > 0 suitably small, where aR , bR are positive constants. Let β1 (t) = u(t)22−θ + ut (t)2θ . Obviously,    f (u), Au =  f  (u), |∇u|2 ≤ C∇u2 + C

 |u|p−1 |∇u|2 dx 

≤ Cu21

p−1 + CuLp+1 ∇u2Lp+1

≤ C(R, ) + C(R)∇u21−δ

φ0 u22 + C(R, ), 8  |π1 | ≤ C(R)u2−θ ut θ u21+θ + u22−θ ≤ C(R)β1 (t)u21+θ , ≤

 φ 0 u22 + ut 21 + C(), |π2 | ≤ g u2 + ut 2−2θ ≤ 8 

 |u| + |u|p |A1−θ ut |dx |π3 | ≤ C

(2.29)



≤ Cuut 2−2θ + CA1−θ ut  ≤ Cuut 1 + Cut 1 u

2N

L N−2(2θ−1)

u

p 2Np

L N+4θ−2

p 2Np

L N+4θ−2

≤ ut 21 + C(R, )u

Using the Hölder inequality with

u

2p

+ C(R, ).

2Np

L N+4θ−2 4θ N+4θ−2

2p 2Np L N+4θ−2

=



+

N−2 N+4θ−2

2Npα

= 1, we obtain that for any α ∈ [0, 1],

2Np(1−α)

|u| N+4θ−2 |u| N+4θ−2 dx

 N+4θ−2 N

(2.30)



≤ u

2pα L

A simple calculation shows that 0 < which implies that

p−1 2p

Npα 2θ

u

2p(1−α) L

2Np(1−α) N−2

.

p−1 2θ(p+1) < min{ 2θ(p+1) Np , 1}. Taking α ∈ ( 2p , min{ Np , 1}),

Npα 2Np(1 − α) (p + 1)N < p + 1, < , 2θ N −2 N −2 we infer from (2.30) that

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Y. Li, Z. Yang / J. Differential Equations ••• (••••) •••–•••

C(R, )u

2p

2pα

2Np

L N+4θ−2

≤ C(R, , ||)uLp+1 u 2pα

≤ C(R, , ||)u1

u

2p(1−α) (p+1)N

L N−2 2p(1−α) L

(p+1)N N−2

(2.31)

2pα(p+1)

≤ C(R, , ||)u12pα−p+1 + C(R, , ||)u ≤ where we have used the fact: u

(p+1)N L N−2

p+1 L

(p+1)N N−2

φ0 u22 + C(R, , ||), 8

p+1 2p(1−α)

p+1

15

> 1, estimate (2.5) and the formula

= |u|  =C

p−1 2

u2

2N L N−2

≤ C|u|

p−1 2

u21 p−1

|u|p−1 |∇u|2 dx ≤ CuLp+1 ∇u21−δ 

≤

φ0 u22 + C(R, , ||). 8C(R, , ||)

Inserting (2.31) into (2.29) gives |π3 | ≤ ut 21 +

φ0 u22 + C(R, ). 8

(2.32)

Inserting (2.29) and (2.32) into (2.27) and making use of (2.28) turn out   d H1 (u, ut ) + κ u22 + ut 21 ≤ C(R)β1 (t)H1 (u, ut ) + C(R) dt

(2.33)

for  > 0 suitably small. 1 When t ∈ (0, 1], multiplying Eq. (2.33) by t 1−θ gives   1  d 1 t 1−θ H1 (u, ut ) + κt 1−θ u22 + ut 21 dt  1 θ  bR t 1−θ u21+θ + ut 21−θ + C(R) ≤C(R)β1 (t)t 1−θ H1 (u, ut ) + 1−θ (2.34)  1 θ  bR 2(1−θ) 2(1−θ) 2θ ≤C(R)β1 (t)t 1−θ H1 (u, ut ) + + C(R) u2θ t 1−θ u1 2 + ut  ut 1 1−θ 

 1 1  ≤C(R)β1 (t)t 1−θ H1 (u, ut ) + κt 1−θ u22 + ut 21 + C(R) 1 + q(θ) , where q(θ) is as shown in (2.9), and where we have used the fact: 2θ−1 θ 1 bR 2(1−θ) ≤ κt 1−θ ut 21 + C(R)q(θ)t θ(1−θ) ut 2 t 1−θ ut 2θ ut 1 1−θ 1

≤ κt 1−θ ut 21 + C(R)q(θ).

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Y. Li, Z. Yang / J. Differential Equations ••• (••••) •••–•••

16

Applying the Gronwall inequality to (2.34) yields  −1 H1 (u(t), ut (t)) ≤ C(R)t 1−θ 1 + q(θ) eC(R) , t ∈ (0, 1].

(2.35)

When t > 1, applying the Gronwall inequality to (2.33) over (1, t) and making use of (2.35) turn out   H1 (u(t), ut (t)) ≤ C(R) 1 + q(θ) + t eC(R)t , ∀t > 1.

(2.36)

Integrating (2.33) over (t, t + 1) and combining the result with (2.35)-(2.36) give

u(t)21+θ

+ ut (t)21−θ

+

t+1 

 u(s)22 + ut (s)21 ds

t

  −1  ≤ C(R) 1 + t 1−θ 1 + q(θ) + t eC(R)t , ∀t > 0.

(2.37)

Formally differentiating Eq. (2.1) with respect to t we get that v = ut solves vtt + φ(u21 )Av + σ (u21 )Aθ vt + f  (u)v + I1 (u, ut ) = 0,

(2.38)

where  I1 (u, ut ) = 2 σ  (u21 )Aθ ut + φ  (u21 )Au (Au, ut ). Taking the multiplier A−θ vt + v in Eq. (2.38) gives

 1 d H2 (v, vt ) + σR −  vt 2 + φ0 v21 ≤ χ1 + χ2 + χ3 , 2 dt

(2.39)

where  H2 (v, vt ) = vt 2−θ + φ(u21 )v21−θ +  2(vt , v) + σ (u21 )v2θ , 

χ1 = −2(Au, v) σ  (u21 )Aθ ut + φ  (u21 )Au, v + A−θ vt ,   χ2 = −(Au, ut ) σ  (u21 )v2θ − φ  (u21 )v21−θ , χ3 = −(f  (u)v, A−θ vt + v). Obviously,  aR (v, vt )2Hθ ≤ H2 (v, vt ) ≤ bR (v, vt )2Hθ  are two positive constants. By virtue of (2.5) and the for  > 0 suitably small, where aR , bR interpolation theorem, we have

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Y. Li, Z. Yang / J. Differential Equations ••• (••••) •••–•••

17



 |χ1 | ≤ C(R)v1 ut θ + u2−θ vθ + vt −θ

 φ0 v21 + C(R, )β1 (t) v2θ + vt 2−θ , 4  |χ2 | ≤ C(R)u2−θ ut θ v2θ + v21−θ ≤

≤ C(R)β1 (t)v2θ , 

 |χ3 | ≤ C (1 + |u|p−1 )|v| |v| + |A−θ vt | dx 





 p−1 ≤ C 1 + uLp+1 vLp+1 vLp+1 + A−θ vt Lp+1 ≤ C(R, )v21−δ + vt 2 ≤

φ0 v21 + vt 2 + C(R, )v2θ . 4

Inserting above estimates into estimate (2.39) and taking  > 0 suitably small, we obtain 

  d H2 (v, vt ) + κ v21 + vt 2 ≤ C(R) 1 + β1 (t) H2 (v, vt ). dt

(2.40)

Multiplying (2.40) by t 2 yields    d2 t H2 (v, vt ) + κt 2 v21 + vt 2 dt    

 v2θ + vt 2−θ ≤C(R) 1 + β1 (t) t 2 H (v, vt ) + 2tbR

  κ ≤C(R) 1 + β1 (t) t 2 H (v, vt ) + t 2 vt 2 + C(R), 2

(2.41)

where we have used the facts:

     2 2tbR v2θ ≤ 2bR + 2bR t ut 2θ v2θ ≤ C(R) + C(R) 1 + β1 (t) t 2 H (v, vt ) , κ   2tbR vt 2−θ ≤ 2tbR vt utt −2θ ≤ C(R) + t 2 vt 2 . 2 Applying the Gronwall lemma to (2.41) and exploiting estimate (2.5), we obtain

ut (t)2θ

+ utt (t)2−θ

t+1

1  + utt (s)2 ds ≤ C(R) + t eC(R)t , ∀t > 0. t

(2.42)

t

The combination of estimates (2.37) and (2.42) yields desired estimate (2.9).

2

By Theorem 2.6, for each θ ∈ [1/2, 1), problem (2.1)-(2.2) generates an evolution semigroup Sθ (t) in phase space H by the formula Sθ (t)(u0 , u1 ) = (uθ (t), uθt (t)), ∀(u0 , u1 ) ∈ H,

(2.43)

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Y. Li, Z. Yang / J. Differential Equations ••• (••••) •••–•••

18

where uθ (t) is the weak solution of problem (2.1)-(2.2) corresponding to initial data (u0 , u1 ) and dissipative index θ ∈ [1/2, 1). We infer from the interpolation theorem, global regularity estimate (2.9) and stability estimate (2.8) that Sθ (t)ξ1 − Sθ (t)ξ2 2H ≤ C(R, t)Sθ (t)ξ1 − Sθ (t)ξ2 Hθ ≤ C(R, t)ξ1 − ξ2 Hθ ≤ C(R, t)ξ1 − ξ2 H for all t > 0 and ξi ∈ H with ξi H ≤ R. That is, Sθ (t) : H → H is locally 1/2-Hölder continuous. 3. Global attractor and its upper semicontinuity Lemma 3.1. Let Assumption 2.1 be valid. Then for every θ ∈ [1/2, 1), the dynamical system (Sθ (t), H) has a bounded absorbing set Bθ , which is of the following properties: (i) Bθ is bounded in H1+θ , closed in H and semi-invariant, i.e., Sθ (t)Bθ ⊂ Bθ for all t ≥ 0; (ii) for all t ≥ 0 and (u(t), ut (t)) = Sθ (t)ξ0 with ξ0 ∈ Bθ , u(t)21+θ

+ ut (t)2θ

+ utt (t)2−θ

+

t+1 

 u(s)22 + ut (s)22θ + utt (s)2 ds

t

≤ C(R0 )q(θ ),

(3.1)

where q(θ) is as shown in (2.9). Proof. Estimate (2.6) shows that the set B0 = {ξ ∈ H|ξ H ≤ R0 } is a uniformly bounded absorbing set of the family of dynamical system (Sθ (t), H), θ ∈ [1/2, 1), that is, for every bounded subset B of H, there exists a t (B) ≥ 0 such that Sθ (t)B ⊂ B0 for all t ≥ t (B) and θ ∈ [1/2, 1). In particular, there exists a t0 > 0 such that Sθ (t)B0 ⊂ B0 for all t ≥ t0 and θ ∈ [1/2, 1). (i) Let   Bθ := Sθ (t)B0 , ∀θ ∈ [1/2, 1). (3.2) t≥t0 +1

H

Obviously, Sθ (t)Bθ ⊂ Bθ for all t ≥ 0 and Bθ is also an absorbing set of Sθ (t). We see from (2.9) that Sθ (1)B0 2H1+θ = sup Sθ (1)ξ 2H1+θ ≤ C(R0 )q(θ ), ξ ∈B 0

which implies that

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Y. Li, Z. Yang / J. Differential Equations ••• (••••) •••–•••

Bθ 2H1+θ = 

 t≥t0 +1

19

Sθ (t)B0 2H1+θ ≤ Sθ (1)B0 2H1+θ ≤ C(R0 )q(θ ),

(3.3)

i.e., Bθ is a bounded subset in H1+θ . (ii) It follows form estimate (2.9) and formula (3.2) that

u(t)21+θ

+ ut (t)2θ

+ utt (t)2−θ

+

t+1 

 u(s)22 + ut (s)21 + utt (s)2 ds ≤ C(R0 )q(θ )

t

(3.4) for all t ≥ 0 and (u(t), ut (t)) = Sθ (t)ξ0 with ξ0 ∈ Bθ . A simple calculation shows that 2p < 2N when 3 ≤ N ≤ 6, which means that V1+θ → L2p , and 2p ∗ ≤ N−2(1+θ) 2p

2p

uL2p ≤ Cu1+θ ≤ C(R0 )q(θ ). When N ≥ 7, if 2p ≤ theorem, we have 2p

2N N−2(1+θ) ,

uL2p ≤ u

2pα

(p−1)N 1−θ

−

where 2pα =

2N L N−4

u

then (3.5) still holds; if 2p >

2p(1−α) 2N L N−2(1+θ)

2p(1+θ) 1−θ

2pα

≤ Cu2

2p(1−α)

u1+θ

(3.5) 2N N−2(1+θ) ,

by the interpolation



≤ C(R0 )q(θ ) 1 + u22 ,

≤ 2. Therefore,



2p  f (u)2 ≤ C 1 + uL2p ≤ C(R0 )q(θ ) 1 + u22 .

(3.6)

By Eq. (2.1) and estimates (3.4), (3.6), t+1 t+1   utt (s)2 + u(s)22 + f (u)2 + g2 ds ut (s)22θ ds ≤ C(R0 ) t

t

≤ C(R0 )q(θ )

t+1 

 1 + utt (s)2 + u(s)22 + g2 ds ≤ C(R0 )q(θ ), ∀t ≥ 0.

t

(3.7) The combination of (3.4) and (3.7) gives (3.1).

2

Theorem 3.2. Let Assumption 2.1 be valid. Then the dynamical system (Sθ (t), H) has a global attractor Aθ which is bounded in H1+θ for every θ ∈ [1/2, 1). Moreover, Aθ = M+ (N ) with N = {(u, 0) ∈ H|φ(u21 )Au + f (u) = g}, and the global attractor Aθ consists of full trajectories γ = {ξ(t)|t ∈ R} such that lim distH {ξ(t), N } = 0 and

t→+∞

lim distH {ξ(t), N } = 0,

t→−∞

where M+ (N ) is the unstable set emanating from N .

(3.8)

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20

[m1+; v1.304; Prn:3/12/2019; 13:21] P.20 (1-33)

Y. Li, Z. Yang / J. Differential Equations ••• (••••) •••–•••

Proof. Lemma 3.1 shows that for every θ ∈ [1/2, 1), the dynamical system (Sθ (t), H) is dissipative and uniformly compact for Bθ ⊂ H1+θ →→ H. Thus, it has a global attractor Aθ in H, which is bounded in H1+θ and can be represented by Aθ = ω(Bθ ) =

 t≥0 s≥t

 Sθ (s)Bθ H ⊂ Bθ .

(3.9)

Energy identity (2.7) shows that E(u, v) is a strict Lyapunov functional and it is continuous on H. Therefore, the dynamical system (Sθ (t), H) is a gradient system, Aθ = M+ (N ) and every full trajectory γ in Aθ is of the property (3.8) (cf. Theorem 2.4.5 in [9]). 2 Theorem 3.3. Let Assumption 2.1 be valid. Then for any θ0 ∈ [1/2, 1), the global attractor Aθ given by Theorem 3.2 is upper semicontinuous at the point θ0 , i.e., lim distH {Aθ , Aθ0 } = 0 with θ ∈ [1/2, 1).

θ→θ0

(3.10)

In order to prove Theorem 3.3, we first quote a lemma. Lemma 3.4. (Kapitansky-Kostin [13]) Assume that the dynamical system (Sλ (t), X) in a complete metric space X possesses a compact global attractor Aλ for every λ ∈ , and  is a complete metric space. Assume that the following conditions hold: (i) there exists a compact set K ⊂ X such that Aλ ⊂ K for all λ ∈ ; (ii) if λk → λ0 , xk → x0 and xk ∈ Aλk , then Sλk (τ )xk → Sλ0 (τ )x0 for some τ > 0. Then the family of global attractors {Aλ }λ∈ is upper semi-continuous at the point λ0 , i.e., lim distX {Aλ , Aλ0 } = 0.

λ→λ0

Proof of Theorem 3.3. For any θ0 ∈ [1/2, 1), since θ → θ0 , without loss of generality we assume that θ ∈ Iθ0 :=

1 1 + θ   0 Bθ . , and Kθ0 := H 2 2 θ∈Iθ0

It follows from estimate (3.1) and the continuity of q(θ) on Iθ0 that Kθ0 2H1+1/2 = 

 θ∈Iθ0

Bθ 2H1+1/2 ≤ C sup Bθ 2H1+θ ≤ sup C(R0 )q(θ ) ≤ Cθ0 , θ∈Iθ0

θ∈Iθ0

i.e., the set Kθ0 is bounded in H1+1/2 and compact in H for H1+1/2 →→ H. By (3.9), Aθ ⊂ Bθ ⊂ Kθ0 for all θ ∈ Iθ0 . Let (uk (t), ukt (t)) = Sθk (t)ξk with θk ∈ Iθ0 , θk → θ0 , and ξk ∈ Aθk , ξk → ξ0 in H. By Lemma 3.1,

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[m1+; v1.304; Prn:3/12/2019; 13:21] P.21 (1-33)

Y. Li, Z. Yang / J. Differential Equations ••• (••••) •••–•••

u

k

(t)21+θk

+ ukt (t)2θk

+ uktt (t)2−θk

+

21

t+1 

 uk (s)22 + ukt (s)22θk + uktt (s)2 ds ≤ Cθ0

t

(3.11) for all t ≥ 0 and k ≥ 0. Then w = uk − u0 solves wtt +

 

 1 1 k φ + φ 0 Aw+ φ k − φ 0 A(uk + u0 ) + σ 0 Aθk − Aθ0 u0t 2 2   + σ 0 Aθk wt + σ k − σ 0 Aθk ukt + f (uk ) − f (u0 ) = 0,

(3.12)

(w(0), wt (0)) = ξk − ξ0 , where φ k = φ(uk 21 ) and σ k = σ (uk 21 ) for k ≥ 0. By estimate (3.11), we can use the multiplier A−1/2 wt + w in Eq. (3.12) and obtain that      d H3 (w, wt ) +  φ k + φ 0 w21 + 2σ 0 wt 2θk −1/2 +  φ k − φ 0 A(uk + u0 ), w dt =2wt 2 +

6 

(3.13) πj ,

j =1

where  1  H3 (w, wt ) = wt 2−1/2 + φ k + φ 0 w21/2 + 2(wt , w) + σ 0 w2θk , 2 2   k 2  k k  0 2 0 0 2 π1 = φ (u 1 )(Au , ut ) + φ (u 1 )(Au , ut ) w1/2 , π2 = σ  (u0 21 )(Au0 , u0t )w2θk ,

 1 π3 = −2σ 0 (Aθk − Aθ0 )u0t , A− 2 wt + w ,

 1 π4 = −2(σ k − σ 0 ) Aθk ukt , A− 2 wt + w , 

1 π5 = −(φ k − φ 0 ) A(uk + u0 ), A− 2 wt ,

 1 π6 = −2 f (uk ) − f (u0 ), A− 2 wt + w . It follows from estimate (3.11) and Assumption 2.1 that φ k ≥ φ0 (≡ φθ0 ) and σ k ≥ σθ0 , and φ(uk 2 ) + φ  (uk 2 ) + σ (uk 2 ) + σ  (uk 2 ) ≤ Cθ , 0 1 1 1 1 k 2 0 2 k 2 0 2 φ(u  ) − φ(u  ) + σ (u  ) − σ (u  ) ≤ Cθ w1 , ∀t ≥ 0, k ≥ 0. 0 1 1 1 1

(3.14)

Thus, there exist positive constants aθ0 and bθ0 such that   aθ0 w21/2 + wt 2−1/2 ≤ H3 (w, wt ) ≤ bθ0 w21+θ0 + wt 2−1/2 2

(3.15)

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22

for  > 0 suitably small, and   |π1 | + |π2 | ≤ Cθ0 uk 1+θk ukt θk + u0 1+θ0 u0t θ0 w2θk ≤ Cθ0 w21+θ0 ≤ 2

φ0 w21 + C(θ0 , )w21/2 , 8

where we have used the fact: 2 − θk ≤ 1 + θk and estimates (3.11), (3.14). Similarly,

 |π3 | ≤ Cθ0 (Aθk −1/2 − Aθ0 −1/2 )u0t  w1 + wt  1 1 φ0 w21 + wt 2 + C(θ0 , )(Aθk − 2 − Aθ0 − 2 )u0t 2 , 8

 |π4 | + |π5 | ≤ Cθ0 w1 ukt θk (wθk + wt θk −1 ) + (uk 3/2 + u0 3/2 )wt −1/2

≤

  φ0 w21 + wt 2 + C(θ0 , ) w21/2 + wt 2−1/2 , 8 

   1 |π6 | ≤ C 1 + |uk |p−1 + |u0 |p−1 |w| |w| + |A− 2 wt | dx ≤





p−1

p−1

≤ C 1 + uk Lp+1 + u0 Lp+1 ≤



1

w2Lp+1 + A− 2 wt 2Lp+1



  φ0 w21 + wt 2 + C(θ0 , ) w21/2 + wt 2−1/2 , 8

and

 k (φ − φ 0 ) A(uk + u0 ), w ≤ C(θ0 , )w1 uk + u0 3/2 w1/2 ≤

φ0 w21 + C(θ0 , )w21/2 . 8

Inserting above estimates into (3.13), taking  > 0 suitably small and making use of (3.15), we have 1 1 d H3 (w, wt ) ≤ Cθ0 H3 (w, wt ) + Cθ0 (Aθk − 2 − Aθ0 − 2 )u0t 2 , dt t 1 1 2 2 (w(t), wt (t))H1/2 ≤ C(θ0 , t)ξk − ξ0 H + C(θ0 , t) (Aθk − 2 − Aθ0 − 2 )u0t (s)2 ds(3.16)

0

for all t ≥ 0. Obviously, 1

1

(Aθk − 2 − Aθ0 − 2 )u0t (s)2 ≤ Cu0t (s)2θ0 ≤ Cθ0 , a.e. s ∈ [0, t]. By the similar argument as in [27], we have 1

1

lim (Aθk − 2 − Aθ0 − 2 )u0t (s)2 = 0, a.e. s ∈ [0, t].

k→∞

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23

Thus by the Lebesgue dominated convergence theorem, we obtain t lim

k→∞

1

1

(Aθk − 2 − Aθ0 − 2 )u0t (s)2 ds = 0.

(3.17)

0

The combination of (3.11), (3.16)-(3.17) shows that (w(t), wt (t))2H ≤ (w(t), wt (t))H1+1/2 (w(t), wt (t))H1/2 

t 1 1 1 2 ≤ C(θ0 , t) ξk − ξ0 H + (Aθk − 2 − Aθ0 − 2 )u0t (s)2 ds → 0. 0

That is, lim Sθk (t)ξk − Sθ0 (t)ξ0 H = 0, ∀t ≥ 0.

k→∞

Therefore, by Lemma 3.4, the family of attractors Aθ is upper semi-continuous at the point θ0 ∈ [1/2, 1). 2 4. Exponential attractor Theorem 4.1. Let Assumption 2.1 be valid. Then the dynamical system (Sθ (t), H) has an exponential attractor Eθ which is bounded in H1+θ for every θ ∈ [1/2, 1). In order to prove Theorem 4.1, we first give some lemmas. Lemma 4.2. [9] Let V : M → M be a mapping defined on a closed bounded set M of a Banach space X. Assume that there exist a Lipschitz mapping K from M into some Banach space Z and a compact seminorm nZ (·) on Z such that V x − V yX ≤ ηx − yX + nZ (Kx − Ky), ∀x, y ∈ M, where 0 < η < 1 is a constant. Then the discrete dynamical system (V k , M) has an exponential attractor E satisfying distX {V k M, E} ≤ Cγ k , ∀k ≥ 1, for some C > 0 and γ ∈ (η, 1). Moreover, dimf (E, X) ≤ ln mZ

2L  1  −1 K , ln γ −η γ

where LK is the Lipschitz constant for K and mZ (R) is the maximal number of elements zi in the ball {z ∈ Z|zZ ≤ R} possessing the property nZ (zi − zj ) > 1 when i = j .

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Lemma 4.3. (Quasi-stability in weaker space) Let Assumption 2.1 be valid. Then for every θ ∈ [1/2, 1), there exist positive constants Cθ and κ such that Sθ (t)ξ1 − Sθ (t)ξ2 Hθ ≤ Cθ e 2

−κt

t ξ1 − ξ2 Hθ + Cθ

Sθ (s)ξ1 − Sθ (s)ξ2 2L2 ×V

2

−2θ

ds, (4.1)

0

for all ξ1 , ξ2 ∈ Bθ and t ≥ 0. Proof. Let (z(t), zt (t)) = (u1 (t), u1t (t)) − (u2 (t), u2t (t)) = Sθ (t)ξ1 − Sθ (t)ξ2 , ∀t ≥ 0. Then, z solves Eq. (2.21) with (z(0), zt (0)) = ξ1 − ξ2 , the estimate (3.1) holds for ui , and formula (2.23) holds for z. Now, we reestimate the terms Ii (i = 1, · · · , 4) on the right hand side of (2.23). Taking account of φi ≥ φθ (a positive constant depending only on θ ), by using estimate (3.1) and the interpolation, we have 1 

  2 1 2 (φ1 − φ2 ) A(u + u ), z ≤  φ  λu1 21 + (1 − λ)u2 21 dλ A(u1 + u2 ), z 0

φθ z21 + C(θ, )z2 , 8   φθ z21 + C(θ, )z2 , |I1 | + |I2 | ≤ Cθ u1 2−θ u1t θ + u2 2−θ u2t θ z2θ ≤ 8 ≤ C(θ, )u1 + u2 21+θ z21−θ ≤

|I3 | ≤ Cθ z1

2  

  ui 2−θ + uit θ zθ + zt −θ

i=1



φθ z21 + C(θ, ) z2θ + zt 2−θ 8 

φθ z21 + zt 2 + C(θ, ) z2 + zt 2−2θ , ≤ 4 ≤

and   |I4 | ≤ Cθ z21−δ + zt 21−δ−2θ ≤



φθ z21 + zt 2 + C(θ, ) z2 + zt 2−2θ . 8

Inserting above estimates into (2.23) and making use of (2.25), we have

 d H (z, zt ) + κH (z, zt ) ≤ Cθ z2 + zt 2−2θ dt for  > 0 suitably small. Applying the Gronwall lemma to (4.2) yields (4.1).

(4.2) 2

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Proof of Theorem 4.1. For any θ ∈ [1/2, 1), taking T > 0 such that Cθ e−κT < 1/4 (see estimate (4.1)). Let X = Hθ , M = Bθ and Vθ = Sθ (T ) in Lemma 4.2. By Lemma 3.1, Bθ is bounded in H1+θ , closed in Hθ and the mapping Vθ : Bθ → Bθ is Lipschitz continuous in the topology of Hθ (see (2.8)). Let the space Z = {(u, ut ) ∈ L2 (0, T ; Hθ )|utt ∈ L2 (0, T ; V−r )} with r > max{N/2, 3θ } equipped with the norm (u, ut )2Z

T

 = u(t)2θ + ut (t)2−θ + utt (t)2−r dt. 0

Let nZ (u, ut ) = Cθ

 T

 1/2 , (u, ut ) ∈ Z. u(t)2 + ut (t)2−2θ dt

0

Obviously, nZ (·) is a compact seminorm on Z (see Lemma 2.2). Define the mapping Kθ : Bθ → Z, Kθ ξ = Sθ (·)ξ = (u(·), ut (·)), ∀ξ ∈ Bθ , where u(·) means uθ (t), t ∈ [0, T ]. Thus by formula (4.1), Vθ ξ1 − Vθ ξ2 Hθ ≤ ηξ1 − ξ2 Hθ + nZ (Kθ ξ1 − Kθ ξ2 ), ξ1 , ξ2 ∈ Bθ , with η2 = Cθ e−κT < 1/4. Let (z(t), zt (t)) = (u1 (t), u1t (t)) − (u2 (t), u2t (t)) = Sθ (t)ξ1 − Sθ (t)ξ2 with ξi ∈ Bθ , i = 1, 2. It follows from Eq. (2.21) and estimate (2.8) that Kθ ξ1 − Kθ ξ2 2Z

T  = z2θ + zt 2−θ + ztt 2−r dt 0

T  ≤ Cθ z2θ + zt 2−θ + Az2−r + Aθ z2−r + f (u1 ) − f (u2 )2−r dt 0

T  z2θ + zt 2−θ + f (u1 ) − f (u2 )2L1 dt ≤ Cθ 0

T  ≤ Cθ z2θ + zt 2−θ + z21 dt 0

≤ C(θ, T )ξ1 − ξ2 2Hθ ,

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that is, the mapping Kθ is Lipschitz continuous. Therefore, by Lemma 4.2 the discrete dynamical system (Vθk , Bθ ) (equipped with Hθ topology) has an exponential attractor Eθ . Let 

Eθ =

(4.3)

Sθ (t)Eθ .

0≤t≤T

Using the similar argument as in [25], one easily shows that Eθ is just an exponential attractor of the dynamical system (Sθ (t), H), and it is bounded in H1+θ . 2 5. Optimal global and exponential attractors Let (u(t), ut (t)) = Sθ (t)ξ with ξ ∈ Bθ . We have known that estimate (3.1) holds for (u, ut , utt ). In this section, we further give the estimate of utt θ−1 for θ − 1 ≥ −θ , then show that the mapping Sθ (t) : Bθ → H1+θ is 1/2-Hölder continuous when t > 0, and then get the desired results. Lemma 5.1. Let Assumptions 2.1 be valid. Then utt (t)2θ−1 ≤ Cθ t −1 (1 + t)2 , ∀t > 0,

(5.1)

where (u(t), ut (t)) = Sθ (t)ξ with ξ ∈ Bθ and Cθ is a positive constant depending only on θ . Proof. Under Assumptions 2.1, estimate (3.1) holds for u and v = ut solves Eq. (2.38). Using the multiplier Aθ−1 vt in Eq. (2.38) gives  1 d vt 2θ−1 + φ(u21 )v2θ + σθ vt 22θ−1 ≤ J1 + J2 + J3 , 2 dt where we have used the fact: σ (u21 ) ≥ σθ > 0 and where J1 = φ  (u21 )(Au, ut )v2θ ,   

J2 = −2(Au, ut ) σ  (u21 ) Aθ ut , Aθ−1 vt + φ  (u21 ) Au, Aθ−1 vt , 

J3 = − f  (u)v, Aθ−1 vt . Using estimate (3.1), Assumptions 2.1 and the Hölder inequality, we have |J1 | ≤ Cθ u2−θ ut θ ut 2θ ≤ Cθ ,   |J2 | ≤ Cθ u2−θ ut θ ut 2θ−1 + u1 vt 2θ−1 σθ ≤ Cθ vt 2θ −1 ≤ vt 22θ−1 + Cθ , 4 

 |J3 | ≤ C 1 + |u|p−1 |v||Aθ−1 vt |dx 



≤ C 1 + u

p−1 L

N(p−1) 2



v

2N

L N−2

Aθ−1 vt 

2N

L N−2

(5.2)

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p−1  v1 vt 2θ−1 ≤ C 1 + u1 σθ ≤ vt 22θ −1 + Cθ ut 21 , 4 where we have used the fact: V1 → L (5.2) gives

N(p−1) 2

for

N (p−1) 2

<

2N N−2 .

Inserting above estimates into



 d vt 2θ−1 + φ(u21 )v2θ + σθ vt 22θ−1 ≤ Cθ 1 + ut 21 . dt

(5.3)

Multiplying (5.3) by t yields  d t vt 2θ−1 + φ(u21 )v2θ + tσθ vt 22θ−1 dt   ≤ tCθ 1 + ut 21 + vt 2θ−1 + φ(u21 )v2θ   ≤ Cθ (1 + t) 1 + ut 21 + utt 2 .

(5.4)

Integrating (5.4) over (0, t) and making use of (3.1), we have utt (t)2θ−1 ≤ Cθ t −1 (1 + t)

t



 (1 + t)2 1 + ut (s)21 + utt (s)2 ds ≤ Cθ , ∀t > 0. t

2

0

Theorem 5.2. Let Assumptions 2.1 be valid. Then for every θ ∈ [1/2, 1) and t > 0, there exists a positive constant Cθ,t such that Sθ (t)ξ1 − Sθ (t)ξ2 V1+θ ×V1−θ ≤ Cθ,t ξ1 − ξ2 H , 1/2

Sθ (t)ξ1 − Sθ (t)ξ2 H1+θ ≤ Cθ,t ξ1 − ξ2 H , ∀ξ1 , ξ2 ∈ Bθ . Proof. (i) For any θ ∈ [1/2, 1) and ξ1 , ξ2 ∈ Bθ , let (z(t), zt (t)) = (u1 (t), u1t (t)) − (u2 (t), u2t (t)) = Sθ (t)ξ1 − Sθ (t)ξ2 , ∀t ≥ 0. Thus ui satisfies estimates (2.24), (3.1), and z solves Eq. (2.21). By estimate (3.1), we can multiply Eq. (2.21) by zt and obtain   1 d Ij , 2zt 2 + φ12 z21 + σθ zt 2θ ≤ 4 dt 4

j =1

where φ12 = φ1 + φ2 is as shown in (2.21), and 

1   1 2 1 1 φ (u 1 ) Au , ut + φ  (u2 21 ) Au2 , u2t z21 , 2

 1 I2 = − (σ1 − σ2 ) Aθ u1t + Aθ u2t , zt 2 I1 =

(5.5)

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1 I3 = − (φ1 − φ2 ) Au1 + Au2 , zt 2 

 I4 = − f (u1 ) − f (u2 ), zt , and where we have used the fact: σi ≥ σθ , i = 1, 2. Taking account of φi ≥ φθ and using estimates (2.24) and (3.1), we have 2zt 2 + φ12 z21 ∼ zt 2 + z21 , |I1 | ≤ Cθ

2  

 ui 1+θ uit θ z21 ≤ Cθ z21 ,

i=1

|I2 | + |I3 | ≤ Cθ z1

2  

 ui 2 + uit 2θ zt 

i=1

  ≤ Cθ (1 + β2 (t)) z21 + zt 2 , where β2 (t) =

2  

 ui (t)22 + uit (t)22θ .

i=1

By the Hölder inequality and the Sobolev embedding: V1+θ → L |I4 | ≤ C



 1 + |u1 |p−1 + |u2 |p−1 |z||zt |dx

≤ C 1 + u1 

p−1 N(p−1)

+ u2 

p−1 N(p−1)



z

2N

L N−2 L 1+θ L 1+θ 

p−1 p−1 ≤ C 1 + u1 1+θ + u2 1+θ z1 zt θ

≤

, we have





N(p−1) 1+θ

zt 

2N

L N−2θ

σθ zt 2θ + Cθ z21 . 2

Inserting above estimates into (5.5) yields    d 2zt 2 + φ12 z21 + 2σθ zt 2θ ≤ Cθ (1 + β2 (t)) 2zt 2 + φ12 z21 , dt   sup zt (s)2 + z(s)21 ≤ Cθ,t ξ1 − ξ2 2H , ∀t ≥ 0.

(5.6)

0≤s≤t

(ii) Taking the multiplier Az + A1−θ zt in Eq. (2.21) gives  1 d πj , H4 (z, zt ) + φθ z22 + (σθ − )zt 21 ≤ 4 dt 4

j =1

(5.7)

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where H4 (z, zt ) = 2zt 21−θ + φ12 z22−θ + σ12 z21+θ + 4(zt , Az), π1 =

 1   i 2 φ (u 1 )(Aui , uit )z22−θ + σ  (ui 21 )(Aui , uit )z21+θ , 2 2

i=1



1 π2 = − (φ1 − φ2 ) A(u1 + u2 ), Az + A1−θ zt , 2 

1 π3 = − (σ1 − σ2 ) Aθ (u1t + u2t ), Az + A1−θ zt , 2 

 π4 = − f (u1 ) − f (u2 ), Az + A1−θ zt . Since σ12 ≥ σθ > 0, there exist positive constant aθ and bθ such that     aθ z21+θ + zt 21−θ ≤ H4 (z, zt ) ≤ bθ z21+θ + zt 21−θ .

(5.8)

Making use of estimates (2.24), (3.1) and the interpolation, we have |π1 | ≤ Cθ

2  

  ui 1+θ uit 1−θ z22−θ + z21+θ

i=1

φθ z22 + C(θ, )z21 , 8    |π2 | ≤ Cθ z1 u1 2 + u2 2 z2 + zt 1 ≤ Cθ z21+θ ≤

φθ z22 + zt 21 + C(θ, )β2 (t)z21 , 8    |π3 | ≤ Cθ z1 u1t 2θ + u2t 2θ z2 + zt 1 ≤

φθ z22 + zt 21 + C(θ, )β2 (t)z21 . 8

≤ By Assumption 2.1, |π4 | ≤ C



  1 + |u1 |p−1 + |u2 |p−1 |z| |A1−θ zt | + |Az| dx.



When N ≥ 4, the fact p < p ∗ implies that N(p − 1) 2N 2N 2N < and < , 2 N − 2(1 + θ ) 2N − 4 + 2θ − p(N − 2 − 2θ ) N − 4 which means that 2N N(p − 1) 2N < and V2−δ1 → L 2N−4+2θ−p(N−2−2θ) for δ1 : 0 < δ1  1. 2 − δ1 N − 2(1 + θ )

(5.9)

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Then by estimate (5.9) and the Hölder inequality, with 2 − δ1 N − 2(2 − δ1 ) 1 + + = 1, N 2N 2 and (p − 1)(N − 2(1 + θ )) N − 2(2θ − 1) 2N − 4 + 2θ − p(N − 2 − 2θ ) + + = 1, 2N 2N 2N we have

 p−1 p−1 |π4 | ≤ C 1 + u1  N(p−1) + u2  N(p−1) z L

2−δ1

p−1 + C 1 + u1  2N

L N−2(1+θ)

L

2−δ1

+ u2 

p−1 2N L N−2(1+θ)



2N

L N−2(2−δ1 )

A1−θ zt 

Az 2N

L N−2(2θ−1)

z

  p−1 p−1 ≤ Cθ 1 + u1 1+θ + u2 1+θ z2−δ1 z2 + zt 1 z2−δ1

2N

L 2N−4+2θ−p(N−2−2θ)

(5.10)

φθ z22 + zt 21 + C(θ, )z22−δ1 8 φθ ≤ z22 + zt 21 + C(θ, )z21 . 4

≤

When N = 3, V1+θ → Lr for all r ≥ 1 and θ ∈ [1/2, 1), thus by (5.9) and the fact: 2(1 − θ ) ≤ 1 we have



 p−1 p−1 |π4 | ≤ C 1 + u1 L4(p−1) + u2 L4(p−1) zL4 A1−θ zt  + Az



 p−1 p−1 ≤ Cθ 1 + u1 1+θ + u2 1+θ z1+θ zt 1 + z2 (5.11)

φθ z22 + zt 21 + C(θ, )z21+θ 8 φθ ≤ z22 + zt 21 + C(θ, )z21 . 4 ≤

The combination of (5.10) and (5.11) turns out |π4 | ≤

φθ z22 + zt 21 + C(θ, )z21 . 4

Inserting above estimates into (5.7) gives   d H4 (z, zt ) + κ z22 + zt 21 ≤ Cθ (1 + β2 (t))z21 dt 1

(5.12)

for  > 0 suitably small. When 0 < t ≤ 1, multiplying Eq. (5.12) by t 1−θ , similarly to the proof of (2.34) we obtain

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  d 1 t 1−θ H4 (z, zt ) ≤ Cθ (1 + β2 (t)) z21 + zt 2 , dt   −1 H4 (z(t), zt (t)) ≤ Cθ t 1−θ sup z(s)21 + zt (s)2 , t ∈ (0, 1].

(5.13)

0≤s≤t

When t > 1, applying the Gronwall lemma to (5.12) over (1, t) and making use of estimate (5.13), we have   H4 (z(t), zt (t)) ≤ Cθ,t sup z(s)21 + zt (s)2 , t > 1.

(5.14)

0≤s≤t

Therefore, the combination of (5.13)-(5.14) and (5.6), (5.8) yields Sθ (t)ξ1 − Sθ (t)ξ2 2V1+θ ×V1−θ = z(t)21+θ + zt (t)21−θ ≤ Cθ,t ξ1 − ξ2 2H , t > 0.

(5.15)

(iii) For any t > 0, using the multiplier zt in Eq. (2.21) yields 

σ12 zt 2θ = − 2(ztt , zt ) − φ12 (Az, zt ) − (φ1 − φ2 ) A(u1 + u2 ), zt 



− (σ1 − σ2 ) Aθ (u1t + u2t ), zt − 2 f (u1 ) − f (u2 ), zt .

(5.16)

By Lemma 5.2 and estimate (3.1), we have 2|(ztt , zt )| ≤ 2ztt θ−1 zt 1−θ ≤ Cθ,t zt 1−θ , ∀t > 0,   |φ12 (Az, zt )| ≤ Cθ z1+θ zt 1−θ ≤ Cθ z21+θ + zt 21−θ ,   

|(φ1 − φ2 ) A(u1 + u2 ), zt | ≤ Cθ u1 1+θ + u2 1+θ zt 1−θ ≤ Cθ zt 1−θ ,   

|(σ1 − σ2 ) Aθ (u1t + u2t ), zt | ≤ Cθ z1 u1t θ + u2t θ zt θ ≤ Cθ z1 ,  



1 2 1 + |u1 |p−1 + |u2 |p−1 |z||zt |dx 2| f (u ) − f (u ), zt | ≤ 

 p−1 p−1 ≤ 1 + u1  2N(p−1) + u2  2N(p−1) z

2N

L N−2(1+θ) L 2(1+2θ)

p−1 p−1  ≤ C 1 + u1 1+θ + u2 1+θ z1+θ zt θ ≤ σθ zt 2θ + Cθ z21+θ , L

2(1+2θ)

zt 

2N

L N−2θ

2N(p−1)

where we have used the interpolation theorem and the Sobolev embedding: V1+θ → L 2(1+2θ) . Inserting above estimates into (5.16) and making use of (5.15) and the fact: σi ≥ σθ > 0, we have  zt (t)2θ ≤ Cθ,t z(t)1 + zt (t)1−θ + z(t)21+θ + zt (t)21−θ  (5.17) ≤ Cθ,t z(t)1+θ + zt (t)1−θ ≤ Cθ,t ξ1 − ξ2 H . The combination of (5.15) and (5.17) gives the desired conclusion. 2

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Theorem 5.3. Let Assumptions 2.1 be valid. Then for each θ ∈ [1/2, 1), (i) the dynamical system (Sθ (t), H) has an optimal global attractor Aθ , which is compact in H1+θ and attracts every bounded subset B of H in the topology of H1+θ , i.e., lim distH1+θ {Sθ (t)B, Aθ } = 0;

t→+∞

(ii) the dynamical system (Sθ (t), H) has an optimal exponential attractor Eθ∗ , whose compactness, boundedness of the fractional dimension and exponential attractiveness for the bounded subset B of H are all in the topology of H1+θ . Proof. By Theorem 5.2, the mapping Sθ (1) : Bθ ⊂ H → H1+θ is 1/2-Hölder continuous for each θ ∈ [1/2, 1). Therefore, (i) Aθ = Sθ (1)Aθ is a compact set in H1+θ . For every bounded subset B of H, there exists a tB > 0 such that Sθ (t − 1)B ⊂ Bθ ⊂ H1+θ for all t ≥ tB + 1, and thus distH1+θ {Sθ (t)B, Aθ } = distH1+θ {Sθ (1)Sθ (t − 1)B, Sθ (1)Aθ } 1/2  ≤ Cθ distH {Sθ (t − 1)B, Aθ } → 0 as t → +∞. (ii) Let Eθ∗ = Sθ (1)Eθ . It follows from the semi-invariance of Eθ that Sθ (t)Eθ∗ = Sθ (1)Sθ (t)Eθ ⊂ Sθ (1)Eθ = Eθ∗ , ∀t ≥ 0. By the 1/2-Hölder continuity of the mapping Sθ (1), we have that the set Eθ∗ is compact in H1+θ , and dimf (Eθ∗ , H1+θ ) ≤ 2dimf (Eθ , H) < +∞, and distH1+θ {Sθ (t)B, Eθ∗ } = distH1+θ {Sθ (1)Sθ (t − 1)B, Sθ (1)Eθ } 1/2  ≤ Cθ distH {Sθ (t − 1)B, Eθ } ≤ C(B)e−γ t , ∀t ≥ tB + 1, where γ is a positive constant. Therefore, Eθ∗ is the desired optimal exponential attractor. 2 Remark 5.4. Under Assumptions 2.1, dimf (Aθ , H1+θ ) ≤ dimf (Eθ∗ , H1+θ ) < +∞. Acknowledgments The authors thank the reviewer for his/her valuable comments and suggestions which helped improving the original manuscript. The research is supported by National Natural Science Foundation of China (Grant No. 11671367).

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