Global dynamics of a fourth-order parabolic equation describing crystal surface growth ∗

In the field of infinite-dimensional dynamical systems, one of the most important issues is to obtain the existence of global attractors for the semigroups of solutions associated with some concrete partial differential equations. There are many studies on the existence of global attractors for diffusion equations. For the classical results, we refer the reader to [2, 3, 8, 19, 20, 24] and the reference cited therein. The model we studied here arises from the study of molecular beam epitaxy. Suppose that F denotes the incident mass flux out of the molecular beam, the height H(x, t) of the surface above the substrate plane satisfies a continuity equation


Introduction
In the field of infinite-dimensional dynamical systems, one of the most important issues is to obtain the existence of global attractors for the semigroups of solutions associated with some concrete partial differential equations.There are many studies on the existence of global attractors for diffusion equations.For the classical results, we refer the reader to [2,3,8,19,20,24] and the reference cited therein.
The model we studied here arises from the study of molecular beam epitaxy.Suppose that F denotes the incident mass flux out of the molecular beam, the height H(x, t) of the surface above the substrate plane satisfies a continuity equation In general, the systematic current J surface depends on the whole surface configuration.Keeping only the most important terms in a gradient expansion, subtracting the mean height H = F u and using appropriately rescaled units of height, distance and time [18], Eq. ( 1) attains the following dimensionless form: where ∆ 2 u describes relaxation through adatom diffusion driven by the surface free energy [13], ∇ • [f (∇u 2 )∇u] models the nonequilibrium current [10], respectively.Assuming in-plane symmetry, it follows that the nonequilibrium current is (anti)parallel to the local tilt ∇u with a magnitude f (∇u 2 ) depending only on the magnitude of the title.Within a Burton-Cabrera-Frank-type theory [11], for small tilts, the current is proportional to |∇u|, and the opposite limit is proportional to |∇u| −1 .Hence, by the interpolation formula f (s 2 ) = 1/(1 + s 2 ) [9,17], we obtain the following equation: where a and µ are positive constants, Ω ⊂ R 2 is a bounded domain, respectively.
In this paper, we study the global dynamics of solutions to Eq. ( 2), which describes the crystal surface growth.We suppose that Ω = [0, L] × [0, L], where L > 0.Moreover, on the basis of physical considerations, the equation is supplemented by the following boundary conditions: for u and the derivatives of u at least of order 3, and the initial condition Remark 1.Since the derivation procedure of ( 2) is attached to a two-dimensional bounded domain Ω ⊂ R 2 , our study focus on the 2D case, which seems meaningful in physical.
During the past years, many authors have paid much attention to Eq. (2).For example, Rost and Krug [17] studied the unstable epitaxy on singular surfaces using Eq. ( 2) with a prescribed slope-dependent surface current.In the limit of weak desorption, Pierre-Louis et al. [15] derived Eq. ( 2) for a vicinal surface growing in the step flow mode.This limit turned out to be singular, and nonlinearities of arbitrary order need to be taken into account.Recently, Grasselli et al. [7] showed that Eq. ( 2) endowed with no-flux boundary conditions generates a dissipative dynamical system under very general assumptions on ∂Ω on a phase-space of L 2 -type.They proved that the system possesses a global as well as an exponential attractor.In [27], based on the iteration technique for regularity estimates and the classical existence theorem of global attractors, Zhao and Liu proved the existence of global attractor for Eq. ( 2) on some affine space of H k (0 k < +∞) when the initial value belongs to H k space.Zhao et al. [26] also consider the existence and uniqueness of time-periodic generalized solutions for Eq. ( 2) in 1D case.Very recently, Zhao and Cao [25], Duan and Zhao [5] invistigated the optimal control problem for Eq. ( 2) in onedimensional and two-dimensional cases, respectively.There are also some other papers related to the well-posedness of molecular beam epitaxy equations in R N and T N ; we refer the reader to [6,12] and the reference cited therein.
In this paper, we are interested in the existence of global attractors for problem (2)- (4).The outline of this paper is as follows.We begin by giving some preparations and the main results on the existence of global attractor in Section 2.Then, in Section 3, we establish some uniform estimates.In Section 4, we prove the main result.The conclusion of this paper is postponed in the last section.

Preliminaries
The weak formulation of problem (2)-( 4) is obtained by multiplying (2) by a test function v ∈ H 2 per (Ω) and using the Green formula and the boundary condition.We find It is worth pointing out that the total mass of the solution u(x, t) is conserved.Indeed, when we replace v by 1 in (5), we find We assume that the initial function satisfies Ω u 0 (x) dx = 0, then it follows that For convenience, in this section, using the same method as [7], we summarize the result on the existence and uniqueness of global solution for problem (2)-(4).
Remark 2. A mild solution to problem (2)-(4) can also be yielded with initial data u 0 ∈ Ḣ2 per (Ω).There are also some classical results on the mild solution to higher-order parabolic equations in the subcritical case of the scale of Banach spaces embedding into L q (Ω)-spaces; we refer the reader to [4,16] and the reference cited therein.
By virtue of Lemma 1 we define the operator semigroup In what follows, we always assume that {S(t)} t 0 is the semigroup generated by the weak solutions of problem ( 2)-( 4).It is sufficiently to see that the restriction of {S(t)} on the affined space Ḣ1 per (Ω) is a well-defined semigroup.In order to prove the existence of global attractor, we give some definitions and results.Definition 1. (See [2,23].)Let B be a bounded subset of H 4 (Ω).B is said to be a bounded (H 1 , H 4 )-absorbing set for {S(t)} t 0 if for every bounded subset E in H 1 , there exists T > 0 depending on B such that Remark 3. The assumption that the operator S(t) is compact on a separable Banach space for all t > 0 (semigroup of compact operators) is met by large classes of dynamical system of physical interest.Triggiani [21,22] pointed out that parabolic PDE defined on bounded spatial domains represent an important subclass of dynamical systems, whose correspondent semigroups are compact for all t > 0. In this paper, in order to let the proof process more complete and systematic, we also give the definition and proof of compactness for S(t)-Definition 2 and Lemma 9. Definition 3. (See [2,23].)Let A be a subset of H 4 (Ω).A is said to be an (H 1 , H 4 )global attractor if the following conditions are satisfied: Proposition 1. (See [2,23].)Let A be an (H The main result of this article is given by the following theorem, which provides the existence of global attractors of problem (2)-(4).
Theorem 1. Suppose that u 0 ∈ Ḣ1 per (Ω), the coefficient a is sufficiently large, then problem (2)-(4) has a ( Ḣ1 per , Ḣ4 per )-global attractor for the solution u(x, t), which is invariant and compact in Ḣ4 per (Ω) and attracts every bounded subset of Ḣ1 per (Ω) with respect to the norm topology of Ḣ4 per (Ω).
Remark 4. In [27], by using iterative principle and the properties of sectorial operator, the authors established the existence of Here, we only assume the initial data u 0 ∈ H 1 per (Ω), and we prove that problem (2)-( 4) has a global attractor in H 4 per (Ω).Our assumption on the initial data seems more relax than [27].
Lemma 5. Suppose that u 0 ∈ Ḣ1 per (Ω), then for problem (2)-( 4), we have Here, M 3 = M 3 (a) is a positive constant depending on a, T 3 = T 3 (a, R) depends on a and R, where Proof.Acting the Laplace operator on (2), we obtain Equation ( 21) is supplemented with the boundary by the following boundary conditions: for u and the derivatives of u at least of order 2 and 5.
Lemma 6. Suppose that u 0 ∈ Ḣ1 per (Ω), a is sufficiently large, then for problem (2)-( 4), we have Here, M 4 = M 4 (a) is a positive constant depending on a, T 4 = T 4 (a, R) depends on a and R, where https://www.mii.vu.lt/NAProof.Differentiating (2) with respect to the time t, setting v = u t , we deduce that Multiplying (29) by v, integrating the resulting relation over Ω, we derive that Using Poincaré's inequality two times, we have It follows from (30) and the above inequality that where a is sufficiently large, it satisfies C a−9µ 2 /(2a) > 0. Using Gronwall's inequality, we derive that The proof is complete.
Here, M 5 = M 5 (a) is a positive constant depending on a, T The proof is complete.
Lemma 8. Suppose that u 0 ∈ Ḣ1 per (Ω), the coefficient a is sufficiently large, then for problem (2)-(4), we have Here, M 6 = M 6 (a) is a positive constant depending on a, T 6 = T 6 (a, R) depends on a and R, where Proof.For Eq. ( 2), by Lemmas 2-7, we have where M 5 and M 6 are positive constants in Section 2, respectively.Since t n → ∞, there exists N > 0 such that t n T for all n N .Therefore, owning to (34), we arrive at On the basis of the compactness of embedding H Hence, it follows from (35) and Sobolev's embedding theorem [1] that Owning to (34) and (36), we derive that Proof of Theorem 1.Note that {S(t)} t 0 has a ( Ḣ1 per , Ḣ1 per )-global attractor A as mentioned above.By Lemma 8, the bounded set B 2 given by ( 33) is a bounded ( Ḣ1 per , Ḣ4 per )absorbing set for {S(t)} t 0 .In addition, Lemma 9 shows that {S(t)} t 0 is ( Ḣ1 per , Ḣ4 per )asymptotically compact.Then, by Proposition 1, A is actually an ( Ḣ1 per , Ḣ4 per )-global attractor for {S(t)} t 0 .The proof is complete.

Conclusion
The dynamic properties of diffusion equation and diffusion system such as the global asymptotical behaviors of solutions and global attractors are important for the study of diffusion model.In this paper, we show that problem (2)-( 4), which models the crystal surface growth, has a H 4 per -global attractor provided that the initial data u 0 ∈ H 1 per (Ω).The results on the existence of global attractor have an analytical complexity slightly about what material scientists normally encounter, then potentially making the analysis more difficult to interpret for a non-mathematician.We also believe that our approach is more satisfying than multiple numerical simulations because with computed solutions there is always the question of whether all interesting states of the system have been detected.