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Well-posedness and longtime behavior for the modified phase-field crystal equation

We consider a modification of the so-called phase-field crystal (PFC) equation introduced by K.R. Elder et al. This variant has recently been proposed by P. Stefanovic et al. to distinguish between elastic relaxation and diffusion time scales. It consists of adding an inertial term (i.e. a second-order time derivative) into the PFC equation. The mathematical analysis of the resulting equation is more challenging with respect to the PFC equation, even at the well-posedness level. Moreover, its solutions do not regularize in finite time as in the case of PFC equation. Here we analyze the modified PFC (MPFC) equation endowed with periodic boundary conditions. We first prove the existence and uniqueness of a solution with initial data in a bounded energy space. This solution satisfies some uniform dissipative estimates which allow us to study the global longtime behavior of the corresponding dynamical system. In particular, we establish the existence of an exponential attractor. Then we demonstrate that any trajectory originating from the bounded energy phase space does converge to a unique equilibrium. This is done by means of a suitable version of the {\L}ojasiewicz-Simon inequality. A convergence rate estimate is also given.


Introduction
The so-called phase-field crystal (PFC) equation has been recently employed to model and simulate the dynamics of crystalline materials, including crystal growth in a supercooled liquid, dendritic and eutectic solidification, epitaxial growth, and so on (see [7,8], cf. also [35,37]). In the phase-field crystal approach, the number density of atoms is approximated by using a phase function φ. This function tends to minimize the following (dimensionless) free energy functional over a spatial bounded (and usually periodic) domain Q The parameter ǫ is a constant with physical significance proportional to the undercooling, i.e., ǫ ∼ T e − T , T e being the equilibrium temperature at which the phase transition occurs [11]. The evolution of φ is thus governed by a (conserved) gradient flow generated by the Fréchet derivative of E (cf. [7,8]), that is, in dimensionless units, where the spatial coordinates are measured in units proportional to the lattice constant. The PFC equation (1.2) is a sixth-order partial differential equation preserving the total mass, which can be viewed as the analog of the (fourth-order) Swift-Hohenberg equation (cf. [43]). Equation (1.2) describes the microstructure of solid-liquid systems at inter-atomic length scales and provides a possibly accurate way to model crystal dynamics, especially defect dynamics in atomic-scale resolution. For more details about the modeling based on the phase-field crystal approach, we refer to the recent review [12] (see also [11,37]). However, one major disadvantage of equation (1.2) is that it fails to distinguish between the elastic relaxation and diffusion time scales (see, e.g., [7,41]). In order to overcome this difficulty and to incorporate both fast elastic relaxation (e.g., in a rapid quasi-phononic time scale) and slower mass diffusion, a modified phase-field crystal (MPFC) model was recently proposed in [41] (cf. also [19,20,42] and [12, Section 3.1.1.2]): where β is a (positive) relaxation time. On the other hand, in the context of phase field techniques applied to fast phase transitions, equations like (1.3) have been derived in [18] to take large deviations from thermodynamic equilibrium into account. We observe that the presence of the inertial term is not a minor modification from the mathematical viewpoint. 3) has been recently studied from the numerical analysis point of view, in [4,46], while the corresponding literature on the simpler PFC equation (1.2) is more abundant (cf., e.g., [3,5,10,22,31,48]). In particular, the authors derived different types of unconditionally energy stable finite difference schemes based on a suitable convex splitting for the free energy E. On the theoretical side, to the best of our knowledge, the only available results are the existence of a weak solution and of a unique strong solution to the MPFC equation (1.3) up to any positive final time T > 0. They were both proven in [45] by using a time discretization scheme and taking the initial value of φ t equal to zero in order to ensure mass conservation (see Section 2). The authors observe: Because of the presence of the second order temporal derivatives, the establishment of a global strong solution and smooth solution for (1.3) is very subtle (see [45,Section 2.3]). Indeed, the existence issue has been investigated only partially so far. For instance, no existence result is available for the so-called energy solutions (see Definition 2.1 below). Such solutions are natural in the sense that they are only required to have finite free energy and they are more general than the weak ones. Besides, uniqueness has been proven only for strong solutions in [45]. Uniqueness of energy (or weak) solutions is not straightforward because of the spatial regularity gap between φ and φ t . On the other hand, the analysis of the longtime behavior of solutions to (1.3) is a further important issue that has not been explored so far. Due the dissipative nature of (1.3), one would expect to find a global attractor as well as an exponential attractor. Also, the convergence of a trajectory to a single equilibrium turns out to be nontrivial, since the stationary set can have a quite complicated structure (see, e.g., [36] for an analysis of the stationary one-dimensional equation).
The goal of the present contribution is to establish first the well-posedness of (1.3) in the energy space without any restriction on the initial value of φ t . A consequence of this is that the mass is conserved only asymptotically and the corresponding (dissipative) dynamical system is no longer a gradient system (i.e. there is no Lyapunov functional). Well-posedness is proven by combining standard (e.g. a Galerkin-type approximation scheme) and nonstandard arguments. Then, we demonstrate the existence of global and exponential attractors as well as the convergence of single trajectories to single stationary states. Such results are based on the (uniform) precompactness of solutions, which is obtained through a suitable decomposition of the solution (due to the hyperbolic-like nature of the equation, see Section 5). Besides, the related proofs require some care also because the system does not possess a Lyapunov functional. In order to overcome this obstacle, we shall treat the autonomous system (1.3) by using some techniques employed for non-autonomous evolution equations [28][29][30] (see Section 6).
For the sake of convenience, we rewrite (1.3) into the following form Here, Q = (0, 1) n is a unit cube in R n , n ≤ 3, without loss of generality. The MPFC equation (1.4) is considered in the periodic setting and is subject to the initial conditions We note that the choice of periodic boundary conditions is realistic since in the crystalline materials the patterns of the nanostructures statistically repeat throughout the domain, which is much larger than the length-scales of atoms. Nevertheless, our theoretical results also hold for homogeneous Neumann boundary conditions, as well as for mixed periodichomogeneous Neumann boundary conditions.
The plan of this paper goes as follows. In Section 2 we introduce a notion of bounded energy solution for the MPFC equation (1.4) with (1.6). This solution is more general than the weak solution considered in [45] (see Definition 2.1). A somewhat similar situation happens in the case of the modified Cahn-Hilliard equation (see [25][26][27], cf. also [21,51] for the one-dimensional case). However, the latter equation is technically a bit more challenging. For instance, in the present case, the sixth-order spatial operator ensures that the weak solutions are globally bounded in three dimensions. This is not true for the modified Cahn-Hilliard equation. Then, in Section 3, we provide some (formal) a priori global dissipative estimates that will be useful in the sequel (cf. Lemmas 3.1 and 3.2). The well-posedness is proven in Section 4 so that we can define a dissipative semigroup acting on the energy phase space (cf. Theorems 4.1 and 4.2). The existence of the global attractor (Theorem 5.1) as well as of an exponential attractors (Theorem 5.2) are established in Section 5. The final Section 6 is devoted to prove the convergence of energy solutions to single equilibria (Theorem 6.1) by using a suitable adaptation of the Lojasiewicz-Simon inequality (cf. Lemma 6.1).
We conclude by observing that an interesting open issue is the construction of a robust family of exponential attractors with respect to the relaxation time β (see [33,Section 3.3]). More precisely, the goal is to show that the existence of a family of exponential attractors depending on β ≥ 0, which is (Hölder) continuous with respect to β. Such a result essentially says that the non-transient dynamics of the MPFC equation (1.3) is close to the one of the PFC equation (1.2) in a quantitative way. The result can be obtained by using, e.g., the argument devised in [32] for the damped semilinear wave equation. Nonetheless, in the present case, there is an additional difficulty related to the already mentioned high (spatial) regularity gap between φ and φ t . This seems to require the construction of a more regular invariant set with respect to what is needed here (see Proposition 5.1) in order to estimate the energy norm of the difference between the solution to the MPCF and the PFC equations, respectively. The corresponding analysis will be carried out in a forthcoming paper.

Preliminaries
Notations and functional spaces. Let V be a real Banach space with norm · V and denote V * its dual space. By < ·, · > V * ,V we indicate the duality product between V and V * . We denote by L(X; Y ) the space of all bounded linear operators from a Banach space X into a second Banach space Y , and we simply write L(X) = L(X; X). Next, we denote by H m p (Q), m ∈ N, the space of functions that are in H m loc (R n ) and periodic with the period Q. For an arbitrary m ∈ N, H m p (Q) is a Hilbert space for the scalar product (u, v) m = |κ|≤m Q D κ u(x)D κ v(x)dx (κ being a multi-index) and its associated norm u m = (u, u) m . For m = 0, H 0 p (Q) = L 2 p (Q) and the inner product as well as the norm on L 2 p (Q) are simply indicated by (·, ·) and · , respectively. The mean value of any function u ∈ L 2 p (Q) is denoted by u = |Q| −1 Q udx and we set u = u − u . For the sake of simplicity, we assume |Q| = 1 hereafter. The dual space of H m p (Q) is denoted by H −m p (Q), which is equipped with the operator norm given by T −m = sup u m=1, u∈H m p (Q) |T (u)|. For m = 1, we introduce an equivalent and more convenient norm associated with the inner product where ψ u (respectively ψ v ) is the unique solution to the elliptic equation in Q subject to periodic boundary conditions: We denote byḢ m p (Q) = {u ∈ H m p (Q) : u = 0} the Sobolev spaces for functions with zero mean. For any u, v ∈L 2 p (Q), we have Then we observe that is a positive operator so that its powers A s 0 (s ∈ R) are well defined. In particular, for s = −1, We also need to introduce the product spaces endowed with the graph norm. Mass conservation. We recall that an important feature of the (parabolic) PFC equation (1.2) is that it enjoys the mass conservation property, namely, However, the mass conservation may fail for the (hyperbolic) MPFC equation (1.4). Nevertheless, it still obeys a conservation law of different type. To this end, we (formally) integrate (1.4) over Q and using the periodic boundary condition, we find which yields the following conservative property after integrating it with respect to time: be a regular solution to problem (1.4)-(1.6). Then we have For the sake of simplicity, we denote By solving the ODE system (2.1)-(2.2) for ( φ(t) , φ t (t) ) with initial conditions φ(0) = φ 0 and φ t (0) = φ 1 , we have the following explicit expressions of φ and φ t : is a solution to problem (1.4)-(1.6) with initial data φ 1 satisfying the zero-mean assumption φ 1 = 0, then φ t (t) = 0 and, in particular, the mass conservation φ(t) = φ 0 holds for all t ≥ 0 (cf. [45,46]).
Energy dissipation. Another important property of the PFC equation (1.2) is that its total energy E(φ) (cf. (1.1)) is decreasing with respect to time, namely, As far as the MPFC equation (1.4) is concerned, in the special case such that φ 1 = 0, one can introduce the following "pseudo energy" to problem (1.4)-(1.6) (cf. [45]) which is also nonincreasing in time, i.e., We note that the theoretical results in [45] were only obtained under the specific assumption φ 1 = 0 that yields φ 1 = 0. Hereafter, we shall not imposed any restriction on the mean of φ 1 . Therefore, the dissipative property (2.6) no longer holds if φ 1 = 0. In this more general situation, problem (1.4)-(1.6) does not have a Lyapunov function. In order to study the long-time behavior of the system, we need to introduce a modified "pseudo energy" to problem (1.4)-(1.6) instead of (2.5), i.e., It is easy to check that the energy functional E(t) coincides with E(t) (cf. (2.5)) provided that φ t = 0. Notion of solution. In order to distinguish the solutions according to their regularities, we introduce the following terminology (reminiscent of [25], in which the Cahn-Hilliard equation with inertial term was considered). (1) A pair (φ, φ t ) is called an energy solution to problem (1.4)-(1.6) , if and the following relations hold (2.10) (2) We say that an energy solution to problem (1. ). The relation (2.9) and initial conditions (2.10) can correspondingly be interpreted in a stronger sense.
(3) We say that an energy solution to problem (1. ). In this case, φ satisfies the equation Remark 2.2. We remark that the initial conditions in (2.10) make sense, since the regularity of (φ, φ t ) in (2.8) ensures the weak continuity (φ, φ t ) ∈ C w (0, T ; X 0 ). Here, we denote C w (0, T ; X) (X being a Banach space) as the topological vector space of all weakly continuous functions f : [0, T ) → X. Remark 2.3. For the sake of simplicity, in this paper we only treat the nonlinearities f of the physically relevant form (1.5). Moreover, in the subsequent analysis, we do not have to impose the restriction α := 1 − ǫ > 0 as in [45,46,48]. Actually, our results hold for more general (possibly non-convex) nonlinearities. For instance, we can take f satisfying the following assumptions: Only in the final section we shall further require f to be real analytic in order to prove the convergence to single equilibria. Note that the physically relevant nonlinearity f (y) = y 3 + (1 − ǫ)y with ǫ ∈ R also satisfies (H1)-(H3).

A priori dissipative estimates
In this section, we first derive some a priori dissipative estimates for the solutions to problem (1.4)-(1.6). Such estimates will be crucial in the subsequent sections. The following calculations are performed in a formal way. However, they can be justified by working within a suitable Faedo-Galerkin approximation scheme (cf. Section 4 below) and then passing to the limit. From now on, the symbols c and c i (i ∈ N) will denote positive constants depending on f, β, ǫ, |Q|, but independent of the initial datum and of time. Their values are allowed to vary even within the same line. Analogously, Q i : R → R stand for generic nonnegative monotone functions. Capital letters like C or C i will be used to indicate constants that have other dependencies (in most cases, on the initial datum). The symbol c Q will denote some embedding constants depending only on the domain Q.
Proof. We first rewrite (1.4) into the following form Multiplying (3.4) by a small constant η > 0 (to be determined later) and add the resulting equation to (3.3), we obtain that Using integration by parts and the Cauchy-Schwarz inequality, we get Then by the expression (1.5), the Young inequality and the Sobolev embedding It follows from the Cauchy-Schwarz inequality that As a result, for η ∈ (0, 1 2β ), we obtain Next, it follows that By the definition of Y 1 (t) and (3.8), for certain small c 5 > 0, we get Next, for κ > 0 satisfying 4κ| φ 1 | ≤ ηc 3 , using the Young inequality and (3.6), we can see that where c κ depends on κ and Q. Collecting the above estimates together, we infer from inequality (3.5) that where c 7 , c 8 , c 9 may depend on β, η, Q, M and | φ 1 |. Then, by the Young inequality and (3.7), we get d dt which yields that Thus, we can deduce our conclusion from (3.9). The proof is complete.
If the initial data are more regular, higher-order dissipative estimates can be obtained, namely, we have is a regular solution to problem (1.4)-(1.6). Then the following estimate holds: where the positive constants ρ 3 , ρ 4 may depend on β, φ 1 , M and |Q|, but independent of φ 0 3 , φ 1 and time t.
Proof. Testing (3.2) by φ t , φ, respectively, we obtain d dt Multiplying (3.18) by η ′ ∈ (0, 1 2β ) and adding the resulting equation to (3.17), we obtain that d dt The uniform dissipative estimate (3.1) together with the Sobolev embedding H 2 (Q) ֒→ L ∞ (Q) (n ≤ 3) yields the (uniform) global boundedness of φ, that is, As a consequence, thanks to (1.5) and the Young inequality, we have Then, using the Sobolev embeddings once more, we are able to estimate the terms in R 2 : Thus we deduce from the expressions of Y 2 , D 2 and R 2 that the following inequality holds where Q 5 is a continuous monotone function satisfying Q 5 (0) = 0. Applying the Gronwall inequality and the dissipative estimate (3.1), we conclude that which combined with (3.1) easily yields estimate (3.16). The proof is complete.

Existence and uniqueness of energy solution
In this section, we shall establish the existence and uniqueness of energy solutions to problem (1.4)-(1.6). Namely, we prove Moreover, any energy solution satisfies the strong time continuity property as well as the following energy identity, for all s, t ∈ [0, T ] with s < t, Before giving the proof of Theorem 4.1, we first establish an auxiliary result which will be useful to establish the time continuity (4.1) and the energy identity (4.2) for energy solutions.
Let us consider the following linear equation subject to the periodic boundary conditions and the initial conditions Here, Λ is a sufficiently large positive constant and G is a given function satisfying We have to the linear problem (4.3)-(4.4). Moreover, the following energy identity holds Proof. The proof of existence can be done through a density argument combined with the semigroup theory. We first approximate G and the initial data with the following sequences Let us denote by Ψ m = (ψ m , βψ m t ) a possible solution to the following linear Cauchy problem For sufficiently large Λ > 0, it is easy to verify that L is a m-accretive operator on X 1 with D(L) = X 3 . As a consequence, the linear problem We infer from the definition of E 0 that for sufficiently large Λ > 0, there exist c, c ′ > 0 such that As a consequence, it follows from (4.9) that Moreover, since G m = 0 (m ∈ N) by assumption, recalling (2.3) and (2.4), we easily see Thus, taking the supremum with respect to t ∈ [0, T ] in (4.11)-(4.13), we deduce that {ψ m } is a Cauchy sequence with respect to the norm of C([0, T ]; . Note that at this stage we only need G ∈ L 2 (0, T ;Ḣ −1 p (Q)) instead of (4.5). Besides, the convergence of ψ m tt can be proved by comparison in (4.3). Therefore, {ψ m } strongly converges to a (unique) solution ψ fulfilling (4.1) and the equation (4.3) is satisfied in the following sense On account of the above strong convergence, it is not difficult to show that ψ satisfies the energy identity (4.6). Such an identity can be (formally) obtained by multiplying equation Proof of Theorem 4.1. The proof relies on a suitable Faedo-Galerkin approximation scheme. We consider the eigenvalue problem −∆w = λw subject to periodic boundary conditions. It is well known that there exist two sequences {λ n } n=1,2,... and {w n } n=1,2,... such that, for every n ≥ 1, λ n ≥ 0 is an eigenvalue and w n = 0 is a corresponding eigenfunction, the sequence λ n is nondecreasing, tending to infinity as n → +∞, and the sequence {w n } is orthonormal and complete in L 2 p (Q). We notice that λ = 0 is an eigenvalue, whence λ 1 = 0, and that any non-zero constant is an eigenfunction (i.e., w 1 = 1). For every i > 1, w i cannot be a constant and w i = 0, whence λ i = Q |∇w i | 2 dx > 0. Moreover, as w 1 = 1 is a constant and {w n } is orthonormal in L 2 p (Q), we easily deduce that A −1 0 w i = λ −1 i w i for every i > 1.
For any n ≥ 1, we introduce the finite-dimensional space W n = span{w 1 , ..., w n } and Π n the orthogonal projection on W n . It is obvious that W n ⊂ C ∞ (Q). Then we consider the approximate problem (P n ): looking for t n > 0 and ξ i ∈ C 2 ([0, t n ]) such that where both relations are intended as equalities in W n . It is easy to verify that Problem (P n ) admits a unique global solution, which satisfies the energy estimate (cf. (3.1)) uniformly with respect to n and time t. Standard compactness arguments (e.g., the Aubin-Lions lemma) permit to take the limit of (4.14)-(4.15) at least for a subsequence. Thus, as a consequence, we obtain existence of a global energy solution (φ, φ t ) to problem (1.4)-(1.6).
The regularity of energy solution is not sufficient for us to prove the uniqueness directly using the energy method. To this end, we use a non-standard argument developed in [39] (cf. also [26]). Let (φ, φ t ) be any energy solution to problem (1.4)-(1.6) on [0, T ], which is the limit of proper subsequence of the approximate solutions (φ n , φ n t ). Define (Φ n , Φ n t ) = Π n (φ, φ t ). Then consider the projection of equation (1.4) which, written for It is obvious that the conservative relation holds so that v n (t) = v n t (t) = 0, ∀ t ≥ 0. (4.18) Since (v n , v n t ) are regular, we can test (4.16) by A −1 0 v n t . This entails d dt We recall that the L ∞ (0, T ; X 0 )-norm of (Φ n , Φ n t ) and (φ n , φ n t ) are bounded (uniformly with respect to n) as well as (φ, φ t ). Therefore, we have where the constant C in the above estimates is independent of n. Using the Young inequality and the Poincaré inequality (cf. (4.18)), we infer from the above estimates that and integrating with respect to time, we get Thus we infer and by the Gronwall inequality we deduce Therefore, we obtain that for n → +∞ Since we already know that (Φ n , Φ n t ) → (φ, φ t ) by definition, it follows by comparison that (φ n , φ n t ) → (φ, φ t ). Hence, the whole sequence (φ n , φ n t ) converges to (φ, φ t ) as n → +∞. This indeed entails the uniqueness of energy solution (φ, φ t ).
Observe now that the unique energy solution can be seen as a solution to the linear equation (4.3) with where Λ > 0 is sufficiently large. From a well-known embedding theorem due to J. Simon [40] we see that the energy solution φ has the continuity property φ ∈ C([0, T ]; H 2−σ p (Q)) for 0 < σ ≪ 1, which implies that G ∈ C([0, T ];Ḣ −1 p (Q)) (here we have used the Sobolev embedding H 2−σ p (Q) ֒→ L ∞ p (Q) for n ≤ 3 and σ small). Thus, we can conclude from Lemma 4.1 that (4.1) holds. On the other hand, using a similar approximation procedure as in the proof of Lemma 4.1, we also infer that φ fulfills the following energy identity where G is given by (4.24). On the other hand, we have so that (4.2) follows from (4.25). The proof of Theorem 4.1 is complete.
The dynamical system. We can now associate with problem (1.4)-(1.6) a semiflow S = (X 0 , S(t)) where S(t) : X 0 → X 0 is the semigroup defined as follows φ being the energy solution given by Theorem 4.1.
From the energy identity satisfied by the energy solution, we are able to check that S(t) is indeed uniformly Lipschitz continuous on bounded balls of X 0 , for any fixed t ≥ 0. Let us consider two pairs of initial data (φ 0j , φ 1j ) (j = 1, 2) in a bounded set of X 0 and write the corresponding equation forφ = φ 1 − φ 2 , φ j being the solution corresponding to the initial datum (φ 0j , φ 1j ). Then, we can take advantage of the associated energy identity with G : ) as a source term in (4.6) so that On the other hand, we have φ (t) = β φ 1 + φ 0 − β φ 1 e − t β . As a consequence, we obtain Thus, by the Gronwall inequality, (4.10) and the fact φ (t) where L 1 , L 2 are positive constants depending on the X 0 -norms of the initial data as well as on β, ǫ, Q and f . Next, due to the dissipative estimate in Lemma 3.1, we can state some dissipative properties of the dynamical system defined on a suitable phase space. Recalling the conservative property (2.2), we have to work on the following subset of X 0 : . Namely, there exists a constant R 0 independent of the initial data such that, for all bounded set For more regular initial data, for instance, (φ 0 , φ 1 ) ∈ X 1 , we can deduce from Lemma 3.2 and a similar Galerkin approximation scheme the existence and uniqueness of weak solutions to problem (1.4)-(1.6) as well as the existence of an absorbing set. More precisely, the following results hold: Namely, there exists a constant R 1 independent of the initial data such that, for all bounded set

Global and exponential attractors
In this section, we study the (global) longtime behavior of the semiflow S associate with the MPFC equation (1.4)-(1.6).

Global attractor
The existence of the global attractor of problem (1. The basic step in the proof of Theorem 5.1 is to show certain (pre)compactness property of trajectories in the phase space X 0 . We have to overcome the difficulties from the hyperbolic nature of the system. To this end, we establish a proper decomposition of the semigroup S(t) into an uniformly asymptotically stable part and a compact part. We note that this decomposition also entails that the attractor A 0 is bounded in the more regular space X 1 . This fact will be further exploited for constructing an exponential attractor.
Let (φ, φ t ) be the unique energy solution to problem (1.4)-(1.6) given in Theorem 4.1. We split this solution into two parts, namely, where and Here, we set with k > 0 being a sufficiently large constant to be chosen later. In particular, we require that f k (s) is monotone and nondecreasing in R.
where κ > 0 is a small constant.
Proof. For any positive constant k, the existence and uniqueness of a global energy solution (φ d (t), φ d t (t)) to problem (5.1)-(5.2) easily follows through the same argument used to prove Theorem 4.1. Moreover, due to the zero-mean assumption on the initial data (5.2), we conclude that Multiplying (5.7) by η > 0 and adding it to (5.6), we get Recalling (3.6), we take k > 0 sufficiently large and η > 0 sufficiently small such that where κ is a (small) constant independent of φ d and time t. As a result, we have We infer from (5.9) that (5.5) holds. The proof is complete.
Proof. Let the constant k be the one we choose in Lemma 5.1. For the initial data (φ 0 , φ 1 ) belonging to a bounded set in X 0 , it follows from the uniform estimates (3.1) and (5.5) that (φ c , φ c t ) also belongs to a bounded set in X 0 , i.e., To prove the fact that (φ c (t), φ c t (t)) is indeed more regular, we perform some higher-order calculations that can be justified rigorously by working within a proper Galerkin scheme as before.
Testing (5.3) by A 0 φ c t and A 0 φ c , respectively, we get d dt Multiplying (5.15) by η 1 > 0 and adding it to (5.14), we deduce First, for sufficiently large k and small η 1 , we can easily see that where κ ′ > 0 is a small constant. Due to the Sobolev embedding H 2 (Q) ֒→ L ∞ (Q) (n ≤ 3), the remainder term R c 1 can be estimated by using the uniform estimates of (φ, φ t ) and (φ c , φ c t ) in the X 0 -norm (see (3.1) and (5.13)) such that The above estimate combined (5.16) and (5.17) yields that As a result, we find On the other hand, the choice of initial data (5.4) indicates that Y c 1 (0) = 0. Then, from (5.17) and (5.19) we conclude that (5.12) holds. The proof is complete.
Proof of Theorem 5.1. We have shown that a given trajectory originating from X 0 is a sum of an exponentially decaying part and a term that belongs to a closed bounded subset of the more regular space X 1 . Therefore, the trajectory is precompact in X 0 . On the other hand, on account of Theorem 4.2, the semigroup S(t) has a bounded attracting set in X M,M ′ 0 (for any β > 0). Hence, the conclusion of Theorem 5.1 follows from a wellknown abstract result for infinite dimensional dynamical systems (see, e.g., [44, Theorem 1.1]).

Exponential attractors
In what follows, we proceed to prove the existence of an exponential attractor for the semiflow S consisting of energy solutions to problem (1.4)-(1.6). For the importance of this notion the reader is referred to [33] and references therein. More precisely, we will prove admits an exponential attractor M 0 , which is a positively invariant, compact subset of X 0 with finite fractal dimension with respect to the X 0 -metric and bounded in X 1 , such that, for any bounded B ⊂ X M,M ′ 0 there exist K B > 0 and γ B > 0 such that where dist X 0 denotes the Hausdorff semidistance of sets with respect to the X 0 -metric.
We note that by its definition an exponential attractor M 0 contains the global attractor A 0 obtained in Theorem 5.1. As a consequence, we have Step 1. Confining the dynamics on a regular positively invariant set in X 1 . Our previous results yield the following preliminary observations: Proof. The conclusion (i) is a simple consequence of Lemmas 5.1 and 5.2. As far as (ii) is concerned, we first recall Lemma 3.2, which gives a dissipative estimate on B 1 (cf. also Theorem 4.4). This entails the existence of a positively invariant and X 1 -bounded set V 1 , which eventually absorbs any X 1 -bounded set of data. In particular, V 1 absorbs B 1 , and by the definition of B 1 in (i), we arrive at (ii). The proof is complete.
Step 2. Existence of a smoother exponentially attracting set in X 1 . Let us take initial data lying in the (regular and positively invariant) set V 1 constructed in Proposition 5.1. Notice also that it is not restrictive to assume V 1 to be weakly closed in X 1 . We now show the asymptotic smoothing property and Hölder continuity of the semigroup S(t) on V 1 .

D c
Let us set z 2 = (φ 20 , φ 21 ) and S(t)z 2 = (φ 2 (t), φ 2t (t)). Recalling that the initial datum is in V 1 , we have the uniform estimate (cf. (3.16)) which also implies φ 2tt (t) −3 ≤ C. Concerning the second term on the right-hand side of (5.34), we infer that As a consequence, from the above estimates and (5.34) we conclude that where C(t * ) is a constant depending on t * , z 1 X 1 and z 2 X 1 . This ends the proof.
Based on the asymptotic smoothing property (Lemma 5.3) and the Hölder continuity of the semigroup S(t) on V 1 (Lemma 5.4), from the abstract result [9, Proposition 1] we deduce the following Proposition 5.2. There exists a bounded set M 0 ⊂ V 1 , closed and of finite fractal dimension in X 0 , positively invariant for the semigroup S(t), such that, for some γ 0 > 0 and K 0 ≥ 0, there holds Step 3. Enlarging the basin of attraction. In what follows, we aim to show that (5.36) actually holds for any bounded subset B ⊂ X M,M ′ 0 instead of the more regular set V 1 , but with different constants K B and γ B . In other words, we have to prove that the basin of exponential attraction coincides with X M,M ′ 0 (recall (5.20)). For this purpose, we recall the transitivity of exponential attraction (cf. [13,Theorem 5.1]), that is, Lemma 5.5. Let X be a metric space with distance function denoted by dist. S(t) is a semigroup acting on X such that dist(S(t)z 1 , S(t)z 2 ) ≤ C 0 e K 0 t dist(z 1 , z 2 ) for some C, K > 0.

Convergence to equilibria
In this section, we investigate the longtime behavior of a single trajectory (φ, φ t ). More precisely, we show that each (energy) solution does converge to a single equilibrium. The main result is as follows Theorem 6.1. For any initial datum (φ 0 , φ 1 ) ∈ X 0 , the unique (energy) solution φ to problem (1.4)-(1.6) fulfills Here, φ ∞ is a stationary solution to problem (1.4)-(1.6), i.e., a solution to the following elliptic equation subject to periodic boundary conditions with an average constraint: Moreover, the following convergence rate estimates hold for all t ≥ 0, where C is a constant depending on (φ 0 , φ 1 ) X 0 and on the coefficients of the system, while θ ∈ (0, 1 2 ) may depend on φ ∞ .
First, we show the decay property of φ t , the time derivative of the phase-field. Recalling the energy equality (4.2), using the uniform estimate (3.1) and the Sobolev embedding theorem, we have for t ≥ s ≥ 0 As we have seen before, E(t) is bounded from below by a constant (depending on Q). Thus (6.7) yields As a result, the function v(t) := φ t is uniformly Lipschitz continuous in H −4 p (Q). This and (6.8) imply that φ t (t) −4 → 0 as t → +∞. Since the trajectory is precompact in X 0 (recalling Section 5), we then have φ t (t) −1 → 0 as t → +∞. Together with (6.6), we arrive at (6.5). The proof is complete.
Thanks to Proposition 6.1 we can define the ω-limit set of (φ 0 , φ 1 ) as follows Next, we give a characterization of the ω-limit set. For any M ∈ R, we set S M = {ψ : ψ satisfies problem (6.2)}. It is standard to show that the energy functional E(φ) admits at least one minimizer ψ ∈ H 2 p (Q) with ψ = M , which solves (6.2). As a consequence, the set S M is nonempty. Moreover, by standard elliptic estimate combined with a bootstrap argument, we see that the solution to problem (6.2) is indeed smooth. Then we have Proposition 6.2. Let the assumptions of Theorem 6.1 hold. The ω-limit set of (φ 0 , φ 1 ) is nonempty and is given by Moreover, the energy functional E(φ) is constant on ω(φ 0 , φ 1 ).
Proof. For any M ∈ R, we introduce the auxiliary functions f M (y) = f (y + M ) and F M (y) = F (y + M ). (6.10) Setting M = β φ 1 + φ 0 , we note that, for any solution φ to problem (1.4)-(1.6), the following relation holds Then we rewrite equation (1.4) in the following form Testing (6.12) by A −1 φ t , we obtain d dt Using the uniform estimate (3.1), the right-hand side of (6.13) can be estimated as follows As a result, we have d dt β . (6.14) Define Differentiating G with respect to time and recalling (6.12), we get Using the uniform estimate (3.1), the Hölder inequality and Young's inequality, we deduce that Let us introduce the function where ν > 0 is sufficiently small so that C 1 ν ≤ 1 2 . From the above estimates (6.14) and (6.18), it follows that where the term on the right-hand side e Similarly to (6.7), we have Since the trajectory is precompact in X 0 , we can find an unbounded increasing sequence {t n } ∞ n=1 in R + such that φ(t n ) − φ ∞ 2 → 0 as n → +∞, for some φ ∞ ∈ H 2 p (Q). Thus the set ω(φ 0 , φ 1 ) is nonempty. We now show that any possible limit point φ ∞ belongs to the set S M . First, it easily follows from (2.4) that φ ∞ = M . Without loss of generality, we may assume t n+1 ≥ t n +1 for n ∈ N. Integrating (6.20) with respect to time on the interval [t n , t n+1 ], we infer from (6.22) that 1 0 φ t (t n + t) 2 −1 dt tn e − 2s β ds → 0, as n → +∞. (6.23) Then we have φ(t n + t 1 ) − φ(t n + t 2 ) −1 → 0 uniformly for t 1 , t 2 ∈ [0, 1] as n → +∞. Thus, from the definition of t n , we see that for t ∈ [0, 1], φ(t n + t) − φ ∞ 2 → 0. By the Lebesgue dominated convergence theorem, we deduce from (6.23) that Recalling the definition of f M and the fact φ ∞ = M , we easily see that φ ∞ solves the stationary problem (6.2). It follows from (6.7) that E(t) converges to a certain constant E ∞ as t → +∞. Since we have shown the convergence of φ t (cf. (6.5)), we see that As a consequence, E(φ) is constant on ω(φ 0 , φ 1 ). The proof is complete.
We are now able to prove the convergence of φ to a single equilibrium φ ∞ . This is not a trivial issue since the energy functional E(φ) is in general nonconvex (for instance, when ǫ > 0 is large). Therefore we do not expect uniqueness of solutions for the stationary problem (6.2). More precisely, the ω-limit set is a subset of S M , whose structure might be complicated (e.g. it can be a continuum), we do not know whether the phase-field φ will converge or not as time goes to infinity, although the sequential convergence holds due to the precompactness of the trajectory.
As a consequence, we have (recall (6.33)) which gives In order to obtain the decay rate in X 0 norm, we test (3.2) by A −1 0 φ t , A −1 0 (φ − φ ∞ ), respectively. We thus obtain d dt and Multiplying (6.39) by a sufficiently small constant η > 0, adding the resulting inequality to (6.38), using interpolation and Young's inequality, we arrive at where Note that, for properly large C 1 > 0, we have On the other hand, (6.40) gives Combining (6.37) with (6.41), we conclude that (6.3) holds. The proof of Theorem 6.1 is now complete.