Pullback attractor of a non-autonomous order-2γ parabolic equation for an epitaxial thin film growth model

The non-autonomous order-2γ parabolic partial differential equation for an epitaxial thin film growth model with dimension d=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d=3$\end{document} is investigated by the method of uniform estimates. The existence of a pullback attractor for the 3D model is proved for γ>3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma >3$\end{document}.


Introduction
In [1], Duan and Zhao showed the existence of pullback D-attractor for a non-autonomous fourth-order parabolic equation. Inspired by [1], we consider the following nonautonomous order-2γ parabolic equation: where h = h(x, t) is the scaled height of a thin film. Ω = [0, L] 3 ⊆ R 3 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 or g ≡ 0, γ ≥ d ≥ 3 was studied by Fan-Alsaedi-Hayat-Zhou. Fan-Samet-Zhou [3] showed the global well-posedness of weak solutions and the regularity of strong solutions for an epitaxial growth model with g ≡ 0. In this note, we study the order-2γ non-autonomous equation of an epitaxial growth model.
Pullback attractors which form a family of compact sets that is bounded in phase space and has invariability under the dynamics system. And the pullback attractor plays a key role in the larger-time behavior of solutions. Compared with uniform attractors, the existence of pullback attractor is easy to get with the weak assumption of a force term. Since the pullback attractor appeared, it has aroused the interest of lots of authors and also has been made great progress. In [4][5][6][7][8], the pullback attractor was presented and proved. The pullback D-attractor of systems for non-autonomous dynamics has been proposed in [9] and pullback D-attractor of non-linear evolution equation was showed in [10]. In [11], the pullback 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 existence of the pullback D-attractor for non-autonomous equation in H 1 0 was showed by Li 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 pullback attractors. However, the existence of pullback D-attractor for a non-autonomous order-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 growth on the external forcing term g(x, t) and gave the proof of the existence of a modified nonautonomous equation. Inspired by [15], we also assume that the external forcing term g(x, t) satisfies the condition of exponential growth, which we give in Sect. 2. To obtain an appropriate prior condition, we must limit the parameter γ > 3 and use the Sobolev inequality many times.
In this note, the pullback D-attractor of an order-2γ non-autonomous model (1) with γ > 3 is studied. The paper is organized as follows. In Sect. 2, we do some preparatory work and give the main result. In Sect. 3, using the method of uniform estimates, the existence of 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 constant C shows different data in different rows.

Definition 2.3 (See [12]) The family
Next, we show Lemma 2.4 which was given in [12]. It is important for the existence proof of pullback attractor for a non-autonomous model.
For convenience, assume that Assume that there exist β ≥ 0 and 0 ≤ α ≤ ( γ +2 4γ 2 -9γ +6 ) 2 ξ for any t ∈ R, such that where ξ is the first eigenvalue of A = (-) γ . And in this paper, let (-) 1 2 = Λ. Using (3), we can obtain for any t ∈ R By a slight modification of the classical results in the autonomous framework, mainly of the Faedo-Galerkin method [17]. In the following, we obtain the result on the existence and uniqueness of solution for system (1).

By Lemma 2.5, we can see that the solution is continuous with initial condition
We can find that the process h(t, τ ) is a norm-to-weak continuous process in the space H γ 0 (Ω). Next we give main result.
Then there exists a unique pullback D-attractor in space H γ 0 (Ω) which is in control of the process corresponding to system (1).

Proof of main result
In this section, we prove the existence of pullback D-attractors for (1). Firstly, we show uniform estimates of solutions for system (1). It has an important effect on the proof that the model (1) has an absorbing set of {h(t, τ )}.
and t τ e ξ s Λ γ h(s) Proof Multiplying (1) 1 by h and integrating it over Ω, we can deduce that From (9), we can get and Multiplying (11) by e ξ (t-τ ) and integrating it over (τ , t), we have Multiplying (12) by e ξ t and integrating it over (t, τ ), we obtain Similarly, multiplying (10) by e ξ t and integrating it over (t, τ ), using (13), it is easy to get t τ e ξ s ∇h(s) The proof is complete.

Lemma 3.3 Suppose that h
Proof Multiplying (1) 1 by (-) γ h and integrating over Ω, then using the Sobolev inequality we can get that is, and Multiplying (30) by (tτ )e ξ t and integrating it over (t, τ ), we get ds . Hence, It the follows from (8) that On the other hand, by Hölder's inequality and (15), we get Combining (33) and (34) with (32), we complete the proof.
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.