Yang et al. Boundary Value Problems (2020) 2020:79 https://doi.org/10.1186/s13661-020-01375-8

R E S E A R C H Open Access

Pullback attractor of a non-autonomousorder-2γ parabolic equation for an epitaxialthin film growth modelXiaojie Yang1, Hui Liu1* and Chengfeng Sun2

*Correspondence:liuhuinanshi@qfnu.edu.cn1School of Mathematical Sciences,Qufu Normal University, Qufu,P.R. ChinaFull list of author information isavailable at the end of the article

AbstractThe non-autonomous order-2γ parabolic partial differential equation for an epitaxialthin film growth model with dimension d = 3 is investigated by the method ofuniform estimates. The existence of a pullback attractor for the 3D model is proved forγ > 3.

Keywords: Pullback attractor; Asymptotic compactness; Non-autonomous parabolicequation

1 IntroductionIn [1], Duan and Zhao showed the existence of pullback D-attractor for a non-autono-mous fourth-order parabolic equation. Inspired by [1], we consider the following non-autonomous order-2γ parabolic equation:

⎧⎨

⎩

ht + ∇ · (1 – |∇h|γ –2)∇h + (–�)γ h = g(x, t), in Ω × [τ ,∞),

h|t=τ = hτ (x),(1)

where h = h(x, t) is the scaled height of a thin film. Ω = [0, L]3 ⊆R3 is the periodic domain.

γ > 3 is a positive constant.The system (1) is modified by equations in [1] and [2]. In [2], the global well-posedness

of strong solution for a γ -order epitaxial growth model with g ≡ 0, γ > 2 and d = 1, 2 org ≡ 0, γ ≥ d ≥ 3 was studied by Fan–Alsaedi–Hayat–Zhou. Fan–Samet–Zhou [3] showedthe global well-posedness of weak solutions and the regularity of strong solutions for anepitaxial growth model with g ≡ 0. In this note, we study the order-2γ non-autonomousequation of an epitaxial growth model.

Pullback attractors which form a family of compact sets that is bounded in phase spaceand has invariability under the dynamics system. And the pullback attractor plays a keyrole in the larger-time behavior of solutions. Compared with uniform attractors, the exis-tence of pullback attractor is easy to get with the weak assumption of a force term. Since thepullback attractor appeared, it has aroused the interest of lots of authors and also has been

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Yang et al. Boundary Value Problems (2020) 2020:79 Page 2 of 12

made great progress. In [4–8], the pullback attractor was presented and proved. The pull-back D-attractor of systems for non-autonomous dynamics has been proposed in [9] andpullbackD-attractor of non-linear evolution equation was showed in [10]. In [11], the pull-back attractors of the n-dimensional non-autonomous parabolic equation was considered.In [12], the norm-to-weak continuous process has been proposed and the proof of the ex-istence of the pullback D-attractor for non-autonomous equation in H1

0 was showed byLi and Zhong. The continuity of pullback and uniform attractors was studied by Hoanga,Olsonb and Robinson [13]. And in [14], Cheskidov and Kavlie did many studies with pull-back attractors. However, the existence of pullback D-attractor for a non-autonomousorder-2γ parabolic has not been studied for γ > 3 yet.

In this paper, we need to overcome the difficulty of the non-linear term ∇ · (1 –|∇h|γ –2)∇h and (–�)γ h. In [15], Park and Park put the condition of exponential growthon the external forcing term g(x, t) and gave the proof of the existence of a modified non-autonomous equation. Inspired by [15], we also assume that the external forcing termg(x, t) satisfies the condition of exponential growth, which we give in Sect. 2. To obtainan appropriate prior condition, we must limit the parameter γ > 3 and use the Sobolevinequality many times.

In this note, the pullback D-attractor of an order-2γ non-autonomous model (1) withγ > 3 is studied. The paper is organized as follows. In Sect. 2, we do some preparatory workand give the main result. In Sect. 3, using the method of uniform estimates, the existenceof a pullback D-attractor for system (1) is proved for γ > 3.

In the following sections, let (1)1 represent the first equation of (1). And note that con-stant C shows different data in different rows.

2 PreliminariesAssume that X is a complete metric space and {h(t, τ )} = {h(t, τ )|t ≥ τ , τ ∈ R} is a two-parameter family of mappings act on X : h(t, τ ) : X → X, t ≥ τ , τ ∈R.

Definition 2.1 (See [16]) A two-parameter family of mappings {h(t, τ )} is called a norm-to-weak continuous process in X if

(1) h(t, s)h(s, τ ) = h(t, τ ) for all t ≥ s ≥ τ ,(2) h(t, τ ) = Id is the identity operator for all τ ∈R,(3) h(t, τ )xn ⇀ h(t, τ )x, if xn → x in X .Assume B(X) is the collection of bounded subsets for X and A, B ⊂ X. Then we denote

the Kuratowski measure of non-compactness α(B) of B as follows:

α(B) = inf{δ > 0 : B has a finite open cover of sets of diameter ≤ δ}. (2)

Let D be a nonempty class of parameterized sets D = {D(t) : t ∈R} ⊂ B(X).

Definition 2.2 (See [12]) A process {h(t, τ )} is said to be pullback ω-D-limit compact iffor any ε > 0 and D ∈D, there exists a τ0(D, t) ≤ t such that α(

⋃τ≥τ0

h(t, τ )D(τ )) ≤ ε.

Definition 2.3 (See [12]) The family A = {A(t) : t ∈ R} ⊂ B(X) is called a pullback D-attractor for h(t, τ ) if

(1) A(t) is compact for all t ∈R,

Yang et al. Boundary Value Problems (2020) 2020:79 Page 3 of 12

(2) A is invariant, i.e., h(t, τ )A(τ ) = A(t) for all t ≥ τ ,(3) A is pullback D-attracting, i.e.,

limτ→–∞ dist

(h(t, τ )D(τ ), A(t)

)= 0, ∀D ∈D, t ∈R.

(4) If {C(t)}t∈R is another family of closed attracting sets, then A(t) ⊂ C(t) for all t ∈R.

Next, we show Lemma 2.4 which was given in [12]. It is important for the existence proofof pullback attractor for a non-autonomous model.

Lemma 2.4 Suppose {h(t, τ )}τ≤t is a norm-to-weak continuous process such that{h(t, τ )}τ≤t is pullback ω-D-limit compact. If there exists a family of pullback D-absorbingsets {B(t) : t ∈ R} ∈ D, i.e., for any t ∈ R and D ∈ D there is a τ0(D, t) ≤ t such thath(t, τ )D(τ ) ⊂ B(t) for all τ ≤ τ0, then there is a pullback D-attractor {A(t) : t ∈R} and

A(t) =⋂

s≤t

⋃

τ≤sh(t, τ )D(τ ).

For convenience, assume that

g(x, t) ∈ L2loc

(R; L2(Ω)

)and sup

t∈R

∫ t+1

t

∣∣g(x, s)

∣∣2 ds < ∞.

Assume that there exist β ≥ 0 and 0 ≤ α ≤ ( γ +24γ 2–9γ +6 )2ξ for any t ∈R, such that

∥∥g(t)

∥∥2 ≤ βeα|t|, (3)

where ξ is the first eigenvalue of A = (–�)γ . And in this paper, let (–�) 12 = Λ.

Using (3), we can obtain for any t ∈R

G1(t) :=∫ t

–∞eξ s∥∥g(s)

∥∥2 ds < ∞, (4)

G2(t) :=∫ t

–∞

∫ s

–∞eξr∥∥g(r)

∥∥2 dr ds < ∞, (5)

∫ t

–∞e– 2(2γ 2–5γ +2)

γ +2 ξ s[[G1(s)] 4γ 2–9γ +6

γ +2 +[G2(s)

] 4γ 2–9γ +6γ +2

]ds < ∞. (6)

By a slight modification of the classical results in the autonomous framework, mainly ofthe Faedo–Galerkin method [17]. In the following, we obtain the result on the existenceand uniqueness of solution for system (1).

Lemma 2.5 Assume that g(x, t) ∈ L2loc(R; L2(Ω)). There is a unique solution h(x, t) such

that(1) if h0 = hτ ∈ L2(Ω) ⇒ h(x, t) ∈ C0([τ ,∞); L2(Ω));(2) if h0 = hτ ∈ Hγ

0 (Ω) ⇒ h(x, t) ∈ C0([τ ,∞); Hγ0 (Ω)).

Yang et al. Boundary Value Problems (2020) 2020:79 Page 4 of 12

By Lemma 2.5, we can see that the solution is continuous with initial condition hτ in thespace Hγ

0 (Ω). Then we define h(t, τ ) : Hγ0 (Ω) → Hγ

0 (Ω) by h(t, τ )hτ . We can find that theprocess h(t, τ ) is a norm-to-weak continuous process in the space Hγ

0 (Ω).Next we give main result.

Theorem 2.6 Assume that g(x, t) ∈ L2loc(R; L2(Ω)) satisfies (3) and hτ ∈ Hγ

0 (Ω) with γ > 3.Then there exists a unique pullback D-attractor in space Hγ

0 (Ω) which is in control of theprocess corresponding to system (1).

3 Proof of main resultIn this section, we prove the existence of pullback D-attractors for (1). Firstly, we showuniform estimates of solutions for system (1). It has an important effect on the proof thatthe model (1) has an absorbing set of {h(t, τ )}.

Lemma 3.1 Assume that g(x, t) ∈ L2loc(R; L2(Ω)) satisfies (3) and hτ ∈ Hγ

0 (Ω) where γ > 3.Consider the system (1), we have, for all t ≥ τ ,

∥∥h(t)

∥∥2 ≤ e–ξ (t–τ )‖hτ‖2 +

1ξ

e–ξ t∫ t

–∞eξ s∥∥g(s)

∥∥2 ds + C (7)

and∫ t

τ

eξ s∥∥Λγ h(s)∥∥2 ds ≤ [

1 + ξ (t – τ )]eξτ‖hτ‖2 +

1ξ

G1(t) + G2(t) + Ceξ t . (8)

Proof Multiplying (1)1 by h and integrating it over Ω , we can deduce that

12

ddt

∫

Ω

h2(t) dx +∫

Ω

∣∣∇h(t)

∣∣γ dx +

∫

Ω

(Λγ h(t)

)2 dx

=∫

Ω

∣∣∇h(t)

∣∣2 dx +

(g(t), h(t)

)

≤ 12∥∥∇h(t)

∥∥γ

γ+

12ξ–1∥∥g(t)

∥∥2 +

12∥∥Λγ h(t)

∥∥2 + C. (9)

From (9), we can get

ddt

∥∥h(t)

∥∥2 +

∥∥∇h(t)

∥∥γ

γ+

∥∥Λγ h(t)

∥∥2 ≤ 1

ξ

∥∥g(t)

∥∥2 + C (10)

and

ddt

∥∥h(t)

∥∥2 + ξ

∥∥h(t)

∥∥2 ≤ 1

ξ

∥∥g(t)

∥∥2 + C. (11)

Multiplying (11) by eξ (t–τ ) and integrating it over (τ , t), we have

∥∥h(t)

∥∥2 ≤ e–ξ (t–τ )‖hτ‖2 +

1ξ

e–ξ t∫ t

–∞eξ s∥∥g(s)

∥∥2 ds + C. (12)

Multiplying (12) by eξ t and integrating it over (t, τ ), we obtain

∫ t

τ

eξ s∥∥h(s)∥∥2 ds ≤ (t – τ )eξτ‖hτ‖2 +

1ξ

∫ t

–∞

∫ s

–∞eξr∥∥g(r)

∥∥2 dr ds + Ceξ t . (13)

Yang et al. Boundary Value Problems (2020) 2020:79 Page 5 of 12

Similarly, multiplying (10) by eξ t and integrating it over (t, τ ), using (13), it is easy to get∫ t

τ

eξ s∥∥∇h(s)∥∥γ

γds +

∫ t

τ

eξ s∥∥Λγ h(s)∥∥2 ds

≤ eξτ‖hτ‖2 +1ξ

∫ t

τ

eξ s∥∥g(s)∥∥2 ds + ξ

∫ t

τ

eξ s∥∥h(s)∥∥2 ds + Ceξ t

≤ [1 + ξ (t – τ )

]eξτ‖hτ‖2 +

1ξ

∫ t

τ

eξ s∥∥g(s)∥∥2 ds

+∫ t

–∞

∫ s

–∞eξr∥∥g(r)

∥∥2 dr ds + Ceξ t . (14)

The proof is complete. �

Lemma 3.2 Let hτ ∈ Hγ0 (Ω), g(x, t) ∈ L2

loc(R; L2(Ω)) and g(x, t) satisfy (3) with γ > 3. Insystem (1) for any t ≥ τ , we have

∥∥∇h(t)

∥∥2 ≤ C

{

e–ξ (t–τ )[‖∇hτ‖2 + ‖hτ‖2] + e–ξ t[

G1(t) +1ξ

G2(t)]

+ 1}

. (15)

Proof Multiplying (1)1 by �h and integrating it over Ω , we derive that

12

ddt

∥∥∇h(t)

∥∥2 +

∥∥Λγ +1h(t)

∥∥2 +

(∇ · (∣∣∇h(t)∣∣γ –2∇h(t)

),�h(t)

)

=∥∥�h(t)

∥∥2 –

(g(t),�h(t)

). (16)

Now we have

(∇ · (∣∣∇h(t)∣∣γ –2∇h(t)

),�h(t)

)

=∫

Ω

∣∣∇h(t)

∣∣γ –2∣∣�h(t)

∣∣2 dx +

∫

Ω

∇h(t)∇(∣∣∇h(t)

∣∣γ –2)

�h(t) dx

=∫

Ω

∣∣∇h(t)

∣∣γ –2∣∣�h(t)

∣∣2 dx +

γ – 22

∫

Ω

∇h(t)∇(∣∣∇h(t)

∣∣2)∣∣∇h(t)

∣∣γ –4

�h(t) dx

=∫

Ω

∣∣∇h(t)

∣∣γ –2∣∣�h(t)

∣∣2 dx + +(γ – 2)

∫

Ω

( 3∑

i,j=3

hijhihj

)( 3∑

k=1

hkk

)∣∣∇h(t)

∣∣γ –4 dx

=∫

Ω

∣∣∇h(t)

∣∣γ –2∣∣�h(t)

∣∣2 dx + (γ – 2)

∫

Ω

( 3∑

i=1

hiih2i

)( 3∑

k=1

hkk

)∣∣∇h(t)

∣∣γ –4 dx

+ (γ – 2)∫

Ω

( 3∑

i,j=1,i�=j

hijhihj

)( 3∑

k=1

hkk

)∣∣∇h(t)

∣∣γ –4 dx

=∫

Ω

∣∣∇h(t)

∣∣γ –2∣∣�h(t)

∣∣2 dx + (γ – 2)

3∑

i=1

∫

Ω

∣∣∇h(t)

∣∣γ –4h2

i h2ii dx

+ (γ – 2)3∑

i,j=1,i�=j

∫

Ω

∣∣∇h(t)

∣∣γ –4h2

i hiihjj dx

+ (γ – 2)∫

Ω

∣∣∇h(t)

∣∣γ –4

�h(t)

( 3∑

i,j=1,i�=j

hijhihj

)

dx. (17)

Yang et al. Boundary Value Problems (2020) 2020:79 Page 6 of 12

By Young’s inequality, we obtain

(γ – 2)3∑

i,j=1,i�=j

∫

Ω

|∇h|γ –4h2i hiihjj dx

≥ –γ – 2

2

3∑

i,j=1,i�=j

∫

Ω

|∇h|γ –4h2i(h2

ii + h2jj)

dx

= –γ – 2

2

3∑

i=1

∫

Ω

|∇h|γ –4h2i h2

ii dx

–γ – 2

2

3∑

i,j=1,i�=j

∫

Ω

|∇h|γ –4h2i h2

jj dx. (18)

On account of the regularity theorem of elliptic equations, we have

(γ – 2)∫

Ω

|∇h|γ –4�h

( 3∑

i,j=1,i�=j

hijhihj

)

dx

≥ –γ – 2

2

∫

Ω

|∇h|γ –2|�h|2 dx. (19)

Using (17), (18) and (19), we get

(∇ · (∣∣∇h(t)∣∣γ –2∇h(t)

),�h(t)

)

≥∫

Ω

∣∣∇h(t)

∣∣γ –2∣∣�h(t)

∣∣2 dx +

γ – 22

3∑

i=1

∫

Ω

|∇h|γ –4h2i h2

ii dx

–γ – 2

2

∫

Ω

|∇h|γ –2|�h|2 dx ≥ 0. (20)

Using the Sobolev inequality, we deduce

∥∥�h(t)

∥∥ ≤ C

∥∥Λγ +1h(t)

∥∥

2γ +1

∥∥h(t)

∥∥

γ –1γ +1 . (21)

Combining (16), (20) and (21) gives

12

ddt

∥∥∇h(t)

∥∥2 +

∥∥Λγ +1h(t)

∥∥2

≤ ∥∥�h(t)

∥∥2 –

(g(t), h(t)

)

≤ ∥∥�h(t)

∥∥2 +

∥∥g(t)

∥∥∥∥�h(t)

∥∥

≤ C∥∥Λγ +1h(t)

∥∥

4γ +1

∥∥h(t)

∥∥

2(γ –1)γ +1 + C

∥∥g(t)

∥∥∥∥Λγ +1h(t)

∥∥

2γ +1

∥∥h(t)

∥∥

γ –1γ +1

≤ 12∥∥Λγ +1h(t)

∥∥2 + C

(∥∥h(t)

∥∥2 +

∥∥g(t)

∥∥2), (22)

that is,

ddt

∥∥∇h(t)

∥∥2 +

∥∥Λγ +1h(t)

∥∥2 ≤ C

(∥∥h(t)

∥∥2 +

∥∥g(t)

∥∥2) (23)

Yang et al. Boundary Value Problems (2020) 2020:79 Page 7 of 12

and

ddt

∥∥∇h(t)

∥∥2 + ξ

∥∥∇h(t)

∥∥2 ≤ C

(∥∥h(t)

∥∥2 +

∥∥g(t)

∥∥2). (24)

Multiplying (24) by eξ (t–τ ) and integrating over (t, τ ), we derive that

∥∥∇h(t)

∥∥2 ≤ e–ξ (t–τ )‖∇hτ‖2 + C

{

e–ξ t∫ t

τ

eξ s∥∥h(s)∥∥2 ds + e–ξ t

∫ t

τ

eξ s∥∥g(s)∥∥2 ds

}

. (25)

By (13) and (25), we get

∥∥∇h(t)

∥∥2 ≤ C

{

e–ξ (t–τ )[‖∇hτ‖2 + ‖hτ‖2] + e–ξ t[

G1(t) +1ξ

G2(t)]

+ 1}

. (26)

Hence, the proof is complete. �

Lemma 3.3 Suppose that hτ ∈ Hγ0 (Ω), g(x, t) ∈ L2

loc(R; L2(Ω)) and g(x, t) satisfies (3) withγ > 3. Consider the system (1), we have for all t ≥ τ

∥∥Λγ h(t)

∥∥2 ≤ C

{[

1 + (t – τ ) +1

t – τ

]

e–ξ (t–τ )[‖hτ‖2 + ‖hτ‖2(4γ 2–9γ +6)

γ +2

+ ‖∇hτ‖2(4γ 2–9γ +6)

γ +2]

+(

1 +1

t – τ

){

1 + e–ξ t[G1(t) + G2(t)]}

+ e–ξ t∫ t

–∞e– 2(2γ 2–5γ +2)

γ +2 ξ s[[G1(s)] 4γ 2–9γ +6

γ +2

+[G2(s)

] 4γ 2–9γ +6γ +2

]ds

}

. (27)

Proof Multiplying (1)1 by (–�)γ h and integrating over Ω , then using the Sobolev inequal-ity we can get

12

ddt

∫

Ω

(Λγ h(t)

)2 dx +∫

Ω

((–�)γ h(t)

)2 dx

=∥∥Λγ +1h(t)

∥∥2 +

(∇ · (∣∣∇h(t)∣∣γ –2∇h(t)

), (–�)γ h(t)

)+

(g(t), (–�)γ h(t)

)

≤ 18∥∥(–�)γ h(t)

∥∥2 +

∥∥Λγ +1h(t)

∥∥2 + C

(∥∥∇ · (∣∣∇h(t)

∣∣γ –2∇h(t)

)∥∥2 +

∥∥g(t)

∥∥2)

≤ 18∥∥(–�)γ h(t)

∥∥2 + C

(∥∥(–�)γ h(t)

∥∥

2γ∥∥Λγ h(t)

∥∥

2(γ –1)γ +

∥∥g(t)

∥∥2

+∥∥∣∣∇h(t)

∣∣γ –2

�h(t)∥∥2)

≤ 14∥∥(–�)γ h(t)

∥∥2 + C

(∥∥Λγ h(t)

∥∥2 +

∥∥g(t)

∥∥2 +

∥∥∇h(t)

∥∥2(γ –2)

2(γ –2)

∥∥�h(t)

∥∥2

∞)

≤ 14∥∥(–�)γ h(t)

∥∥2 + C

(∥∥Λγ h(t)

∥∥2 +

∥∥g(t)

∥∥2

+∥∥(–�)γ h(t)

∥∥

3(γ –3)2γ –1

∥∥∇h(t)

∥∥

4γ 2–13γ +132γ –1

∥∥(–�)γ h(t)

∥∥

52γ –1

∥∥∇h(t)

∥∥

4γ –72γ –1

)

≤ 12∥∥(–�)γ h(t)

∥∥2 + C

(∥∥Λγ h(t)

∥∥2 +

∥∥g(t)

∥∥2 +

∥∥∇h(t)

∥∥

2(4γ 2–9γ +6)γ +2

), (28)

Yang et al. Boundary Value Problems (2020) 2020:79 Page 8 of 12

that is,

ddt

∥∥Λγ h(t)

∥∥2 +

∥∥(–�)γ h(t)

∥∥2 ≤ C

(∥∥Λγ h(t)

∥∥2 +

∥∥g(t)

∥∥2 +

∥∥∇h(t)

∥∥

2(4γ 2–9γ +6)γ +2

)(29)

and

ddt

∥∥Λγ h(t)

∥∥2 + ξ

∥∥Λγ h(t)

∥∥2 ≤ C

(∥∥Λγ h(t)

∥∥2 +

∥∥g(t)

∥∥2 +

∥∥∇h(t)

∥∥

2(4γ 2–9γ +6)γ +2

). (30)

Multiplying (30) by (t – τ )eξ t and integrating it over (t, τ ), we get

(t – τ )eξ t∥∥Λγ h(t)∥∥2

≤ C{∫ t

τ

[1 + (s – τ )

]eξ s∥∥Λγ h(s)

∥∥2 ds +

∫ t

τ

(s – τ )eξ s∥∥g(s)∥∥2 ds

+∫ t

τ

(s – τ )eξ s∥∥∇h(s)∥∥

2(4γ 2–9γ +6)γ +2 ds

}

. (31)

Hence,

∥∥Λγ h(t)

∥∥2 ≤ C

(

1 +1

t – τ

)

e–ξ t∫ t

τ

eξ s∥∥Λγ h(s)∥∥2 ds + Ce–ξ tG1(t)

+ Ce–ξ t∫ t

τ

eξ s∥∥∇h(s)∥∥

2(4γ 2–9γ +6)γ +2 ds

=: I1 + I2 + I3. (32)

It the follows from (8) that

I1 ≤ C(

1 +1

t – τ

)

e–ξ t{[1 + ξ (t – τ )

]eξτ‖hτ‖2 +

1ξ

G1(t) + G2(t) + Ceξ t}

≤ C[

1 + (t – τ ) +1

t – τ

]

e–ξ (t–τ )‖hτ‖2 + C(

1 +1

t – τ

)

e–ξ t[G1(t)

+ G2(t)]

+ C(

1 +1

t – τ

)

. (33)

On the other hand, by Hölder’s inequality and (15), we get

I3 ≤ Ce–ξ t∫ t

τ

eξ s{e–ξ (s–τ )(‖∇hτ‖2 + ‖hτ‖2)

+ e–ξ s[G1(s) + G2(s)]

+ 1} 4γ 2–9γ +6

γ +2 ds

≤ Ce–ξ t∫ t

τ

eξ s ds + ce–ξ t∫ t

τ

eξ se– 4γ 2–9γ +6γ +2 ξ (s–τ )(‖hτ‖2

+ ‖∇hτ‖2)4γ 2–9γ +6

γ +2 ds

+ Ce–ξ t∫ t

τ

eξ se– 4γ 2–9γ +6γ +2 ξ s{[G1(s)

] 4γ 2–9γ +6γ +2 +

[G2(s)

] 4γ 2–9γ +6γ +2

}ds

≤ Ce–ξ t[eξ t – eξτ]

+ ce–ξ (t–τ )(‖hτ‖2(4γ 2–9γ +6)

γ +2

Yang et al. Boundary Value Problems (2020) 2020:79 Page 9 of 12

+ ‖∇hτ‖2(4γ 2–9γ +6)

γ +2)∫ t

τ

e– 4γ 2–9γ +6γ +2 ξ (s–τ ) ds

+ Ce–ξ t∫ t

τ

e– 2(2γ 2–5γ +2)γ +2 ξ s{[G1(s)

] 4γ 2–9γ +6γ +2 +

[G2(s)

] 4γ 2–9γ +6γ +2

}ds

≤ C + C(t – τ )e–ξ (t–τ )(‖hτ‖2(4γ 2–9γ +6)

γ +2 + ‖∇hτ‖2(4γ 2–9γ +6)

γ +2)

+ Ce–ξ t∫ t

τ

e– 2(2γ 2–5γ +2)γ +2 ξ s{[G1(s)

] 4γ 2–9γ +6γ +2 +

[G2(s)

] 4γ 2–9γ +6γ +2

}ds. (34)

Combining (33) and (34) with (32), we complete the proof. �

Remark 3.4 By Sobolev inequality and (28), we can get the exponent of ‖(–�)γ h(t)‖ is3(γ –3)2γ –1 > 0, which is in the seventh row in (28). Hence, we limit γ > 3 with d = 3. Inspired

by [12] and [15], the existence of pullback D-attractor for system (1) is proved as follows.Firstly, assume that R is the set of all function r : R → (0,∞) such that

limt→∞ teδtr

2(4γ 2–9γ +6)γ +2 (t) = 0.

Let D := {D(t) : t ∈ R} ⊂ B(Hγ0 (Ω)), then D(t) ⊂ B0(r(t)) for some r(t) ∈R. Here B0(r(t)) ⊆

Hγ0 (Ω) is a closed sphere and its radius is r(t). Assume

r20(t) = 2C

{

1 + e–ξ tG1(t) + e–ξ tG2(t)

+ e–ξ t∫ t

–∞e– 2(2γ 2–5γ +2)

γ +2 ξ s[[G1(s)] 4γ 2–9γ +6

γ +2

+[G2(s)

] 4γ 2–9γ +6γ +2

]ds

}

. (35)

Using the continuity of embedding Hγ0 (Ω) ↪→ L2(Ω), for any D ∈ D and t ∈ R there

exists τ0(D, t) < t, such that

∥∥Λγ h(t)

∥∥ ≤ r0(t), ∀τ < τ0. (36)

In addition, since 0 ≤ α ≤ ( γ +24γ 2–9γ +6 )2ξ , we have B0(r0(t)) ∈ D. Hence, by Lemma 2.4,

there is a family of pullback D-absorbing sets B0(r0(t)) in Hγ0 (Ω).

Then the main idea is proved.

Proof of Theorem 2.6 By Lemma 2.4, we need only to demonstrate that the process h(t, τ )is pullback ω-D-limit compact. Then we can obtain the existence of pullback D-attractor.The operator A–1 ∈ L2(Ω) is continuous and compact, so there is a sequence {ξj}∞j=1 satis-fying

ξ1 ≤ ξ2 ≤ · · · ≤ ξj ≤ · · · , ξj → ∞, as j → ∞,

and there is a family {wj}∞j=1, where wj ∈ Hγ0 (Ω), j = 1, 2, . . . . The elements in {wj}∞j=1 are

orthonormal in L2(Ω) such that

Awj = ξjwj, for j = 1, 2, . . . .

Yang et al. Boundary Value Problems (2020) 2020:79 Page 10 of 12

Let Xn = span{w1, w2, . . . , wn} ⊂ Hγ0 (Ω) and Pn : Hγ

0 (Ω) → Xn be an orthogonal projec-tor. Therefore

h = Pnh + (I – Pn)h := h1 + h2. (37)

Testing (1)1 by (–�)γ h2, using Young’s inequality, we get

12

ddt

∥∥Λγ h2(t)

∥∥2 +

∥∥(–�)γ h2(t)

∥∥2

≤ 12∥∥(–�)γ h2(t)

∥∥2 + C

(∥∥Λγ h(t)

∥∥2 +

∥∥g(t)

∥∥2 +

∥∥∇h(t)

∥∥

2(4γ 2–9γ +6)γ +2

), (38)

which means

ddt

∥∥Λγ h2(t)

∥∥2 + ξn

∥∥Λγ h2(t)

∥∥2 ≤ C

(∥∥Λγ h(t)

∥∥2 +

∥∥g(t)

∥∥2 +

∥∥∇h(t)

∥∥

2(4γ 2–9γ +6)γ +2

). (39)

Multiplying (39) by (t – τ )eξnt and integrating it over (τ , t), we obtain

(t – τ )eξnt∥∥Λγ h2(t)∥∥2

≤∫ t

τ

eξns∥∥Λγ h2(s)∥∥2 ds + C

∫ t

τ

(s – τ )eξns∥∥Λγ h(s)∥∥2 ds

+ C∫ t

τ

(s – τ )eξns∥∥g(s)∥∥2 ds + C

∫ t

τ

(s – τ )eξns∥∥∇h(s)∥∥

2(4γ 2–9γ +6)γ +2 ds

≤∫ t

τ

eξns∥∥Λγ h(s)∥∥2 ds + C(t – τ )

{∫ t

τ

eξns∥∥Λγ h(s)∥∥2 ds

+∫ t

τ

eξns∥∥g(s)∥∥2 ds +

∫ t

τ

eξns∥∥∇h(s)∥∥

2(4γ 2–9γ +6)γ +2 ds

}

. (40)

Combining (36) and (40) gives for any τ ≤ τ0

∥∥Λγ h2(t)

∥∥2

≤ (t – τ )–1e–ξnt∫ t

τ

eξns∥∥Λγ h(s)∥∥2 ds + Ce–ξnt

∫ t

τ

eξns∥∥Λγ h(s)∥∥2 ds

+ Ce–ξnt∫ t

τ

eξns∥∥g(s)∥∥2 ds + Ce–ξnt

∫ t

τ

eξns∥∥∇h(s)∥∥

2(4γ 2–9γ +6)γ +2 ds

≤ (t – τ )–1e–ξnt∫ t

τ

eξnsr20(s) ds + Ce–ξnt

∫ t

τ

eξnsr20(s) ds

+ Ce–ξnt∫ t

τ

eξns∥∥g(s)∥∥2 ds + Ce–ξnt

∫ t

τ

eξnsr2(4γ 2–9γ +6)

γ +20 (s) ds. (41)

Note that

e–ξnt∫ t

τ

eξnsr20(s) ds ≤ Ce–ξnt

∫ t

τ

eξns[1 + e–ξ sG1(S) + e–ξ sG2(s)]

ds

≤ Cξ–1n + C(ξn – ξ )–1e–ξ t[G1(t) + G2(t)

](42)

Yang et al. Boundary Value Problems (2020) 2020:79 Page 11 of 12

and

e–ξnt∫ t

τ

eξnsr2(4γ 2–9γ +6)

γ +20 (s) ds

= Ce–ξnt∫ t

τ

eξns{

1 + e–ξ sG1(s) + e–ξ sG2(s)

+ e–ξ s∫ s

–∞e– 2(2γ 2–5γ +2)

γ +2 ξr[[G1(r)] 4γ 2–9γ +6

γ +2 +[G2(r)

] 4γ 2–9γ +6γ +2

]dr

} 4γ 2–9γ +6γ +2

ds

≤ Cξ–1n

+ C(

ξn –4γ 2 – 9γ + 6

γ + 2ξ

)–1

e– 4γ 2–9γ +6γ +2

{[G1(t)

] 4γ 2–9γ +6γ +2 +

[G2(t)

] 4γ 2–9γ +6γ +2

}

+ C(

ξn –4γ 2 – 9γ + 6

γ + 2ξ

)–1

e– 4γ 2–9γ +6γ +2

·{(∫ t

–∞e– 2(2γ 2–5γ +2)

γ +2 ξ s[G1(s)] 4γ 2–9γ +6

γ +2 ds) 4γ 2–9γ +6

γ +2

+(∫ t

–∞e– 2(2γ 2–5γ +2)

γ +2 ξ s[G1(s)] 4γ 2–9γ +6

γ +2 ds) 4γ 2–9γ +6

γ +2}

. (43)

On the other hand, by simple calculations, we get

e–ξnt∫ t

τ

eξns∥∥g(s)∥∥2 ds ≤

⎧⎨

⎩

βe–ξnt ∫ tτ

eξnse–αs ds, t ≤ 0,

βe–ξnt ∫ tτ

eξnseα|s| ds, t ≥ 0

≤⎧⎨

⎩

βe–αt

ξn–α, t ≤ 0,

βeαt

ξn+α+ βe–ξnt

ξn–α, t ≥ 0.

(44)

By (41), (42), (43) and (44), we see that, for any ε > 0, there exist τ0 < t and N ∈ N suchthat for every n > N and every τ < τ0

∥∥Λγ h2(t)

∥∥ ≤ ε.

It shows that the process {h(t, τ )} is pullback ω-D-limit compact.Then, using Lemma 2.4, the proof of Theorem 2.6 is complete. �

AcknowledgementsThe authors really appreciate the anonymous reviewers for their pertinent comments and suggestions, which werehelpful to improve the earlier manuscript.

FundingThe second author is supported by the Natural Science Foundation of Shandong under Grant No. ZR2018QA002, theNational Natural Science Foundation of China No. 11901342 and China Postdoctoral Science FoundationNo. 2019M652350. The third author is supported by the NSF of China (No. 11701269).

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors read and approved the final manuscript.

Yang et al. Boundary Value Problems (2020) 2020:79 Page 12 of 12

Author details1School of Mathematical Sciences, Qufu Normal University, Qufu, P.R. China. 2School of Applied Mathematics, NanjingUniversity of Finance and Economics, Nanjing, P.R. China.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 18 January 2020 Accepted: 7 April 2020

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