Longtime behavior of nonlocal Cahn-Hilliard equations

Here we consider the nonlocal Cahn-Hilliard equation with constant mobility in a bounded domain. We prove that the associated dynamical system has an exponential attractor, provided that the potential is regular. In order to do that a crucial step is showing the eventual boundedness of the order parameter uniformly with respect to the initial datum. This is obtained through an Alikakos-Moser type argument. We establish a similar result for the viscous nonlocal Cahn-Hilliard equation with singular (e.g., logarithmic) potential. In this case the validity of the so-called separation property is crucial. We also discuss the convergence of a solution to a single stationary state. The separation property in the nonviscous case is known to hold when the mobility degenerates at the pure phases in a proper way and the potential is of logarithmic type. Thus, the existence of an exponential attractor can be proven in this case as well.


Introduction
The Cahn-Hilliard equation was proposed in [7] as a model for (isothermal) phase separation phenomena in binary alloys. Since then it was analyzed by many authors and used in several different contexts (see, e.g., [9,40] and references therein). The basic form of such an equation is the following where ϕ is the relative difference of the two phases (or the concentration of one phase), and µ is the so-called chemical potential given by in Ω×(0, ∞) , where Ω ⊂ R d , d = 2, 3, is a bounded domain with Lipschitz boundary Γ = ∂Ω. Here κ is the mobility coefficient, ǫ > 0 is a given (small) parameter related to the thickness of the interface separating the two phases, and F is the (density) of potential energy. A physically relevant choice for F is the following In the literature, it is common to distinguish between singular potentials, which are defined on finite intervals like (1.3), and regular ones as (1.4), defined on R . We recall that equations (1.1)-(1.2) have been deduced phenomenologically, i.e., as the (conserved) gradient flow associated with the Fréchet derivative of the free energy functional In [29,30], starting from a microscopic model, the authors rigorously derived a macroscopic equation for phase segregation phenomena. This is a nonlocal version of the Cahn-Hilliard equation, namely, the chemical potential is given by where J ǫ (x) = ǫ −d J(ǫ −1 x). By using formal asymptotic analysis, the authors also showed that the interface evolution problems associated with such equation as ǫ goes to 0 are exactly the ones associated with the standard Cahn-Hilliard equation (i.e., Stefan-like and Mullins-Sekerka problems). In addition, also the nonlocal version can be viewed as the conserved gradient flow associated with the first variation of the free energy functional As a consequence, we can observe (formally) that the nonlocal interaction term can be locally approximated by the square gradient, provided that J is sufficiently concentrated around 0. That is, the functional L can be viewed as a local approximation of N . This was already noted by Van der Waals (see [41]). Thus the nonlocal Cahn-Hilliard equation seems well justified and more general than the classical one, though the related literature is far less abundant. In particular, most of the theoretical results are devoted to well-posedness, but very few are concerned with the longtime behavior of solutions. The main reason is related to the eventual boundedness and regularization of the order parameter which are needed to prove the precompactness of trajectories in some convenient topology. Well-posedness and regularity issues were firstly analyzed in [30] on a three-dimensional torus with degenerate mobility and logarithmic potential. A similar equation endowed with no flux boundary condition was studied in [23] (cf. also [11,20,21] and, for viscous versions, [33,34]). For this case, the convergence to a single stationary state of a given trajectory was proven in [35] through a suitable Lojasiewicz-Simon inequality. This fact required to show preliminarily that a solution stays eventually strictly away from the pure phases: the so-called separation property.
For the constant mobility case and regular potentials, some existence, uniqueness and regularity results were obtained in [5] (see also [6,32]). In that paper the existence of bounded absorbing sets was also established. Nevertheless, no results were known about the existence of more interesting invariant objects like, e.g., global attractors (cf. [38] and its references). Only recently, the existence of a (connected) global attractor has been proven in [18] for constant mobility and regular potentials (see [19] for singular ones). This has been done by exploiting the energy identity as a by-product of a result related to a more complicated model for phase separation in binary fluids. A natural question now arises: does the global attractor have finite (fractal) dimension? Here we give a positive answer and we actually prove more, namely, the existence of an exponential attractor (see again [38] for details). More precisely, taking for simplicity κ = ǫ = 1, we consider the following nonlocal Cahn-Hilliard equation in Ω × (0, ∞) , (1.6) in Ω × (0, ∞) , (1. 7) subject to the no-flux boundary condition (1.8) ∂ n µ = 0, on Γ × (0, ∞) and to the initial condition (1.9) ϕ(0) = ϕ 0 , in Ω.
Here the coefficient α ≥ 0 characterizes the possible influences of internal microforces (see, e.g., [39]). The presence of this term is not necessary in the case of regular potentials, while it is crucial in the case of singular ones. In fact, in the former case, in order to prove our main result we need to first establish the eventual boundedness of ϕ. This boundedness is, say, built-in in the latter case, but we need to show that ϕ has the separation property uniformly with respect to the initial data. This feature is an open problem even for the classical local Cahn-Hilliard equation with constant mobility in dimension three (see [37]). The paper is organized as follows. Section 2 is devoted to the nonviscous case with a regular potential, while Section 3 is concerned with the viscous equation with a singular potential. Provided suitable global bounds are obtained (this is the most technical part), the existence of an exponential attractor is proven through a short trajectory type technique devised in [15]. We also show that, in both cases, each solution converges to a single equilibrium by using a suitable version of the Lojasiewicz-Simon inequality, provided that F is real analytic. In the final Section 4, we consider the (nonviscous) equation with degenerate mobility and logarithmic potential. On account of the validity of the separation property, we can still prove the existence of an exponential attractor.
2. The nonviscous case with regular potential 2.1. Some preliminary results. We begin with some basic notation and assumptions. Let us first set H := L 2 (Ω) and V := H 1 (Ω) . For every ψ ∈ V ′ , V ′ the dual space of V , we denote by ψ the average of ψ over Ω, that is, where |Ω| stands for the Lebesgue measure of Ω and ·, · is the duality product. Then we introduce the spaces With these definitions, it is well known that A N |V0 maps V 0 into V ′ 0 isomorphically, and that the inverse map These maps also satisfy the following well-known relations: The assumptions listed below are the same as in [5] (see also [10]). ( in Ω. (H2) F ∈ C 2,1 loc (R) and there exists c 0 > 0 such that (H3) There exist c 1 > 1 2 J L 1 (R d ) and c 2 ∈ R such that F (s) ≥ c 1 s 2 − c 2 , ∀s ∈ R.
(H4) There exist c 3 > 0, c 4 ≥ 0 and p ∈ (1, 2] such that (H5) F ∈ C 2 (R) and there exist c 5 , c 6 > 0 and q > 0 such that Remark 2.1. Note that the operator ψ → J * ψ is self-adjoint and compact from H to itself, provided that (H1) is satisfied. Also, it is easy to realize that it is compact from L ∞ (Ω) to C 0 (Ω) and that a ∈ L ∞ (Ω).
The next result can also be found in [18,Theorem 1].
Proposition 2.5. Let m ≥ 0 be given. Then every weak solution to (1.6)-(1.9) satisfies the dissipative estimate: provided that | ϕ 0 | ≤ m, where k and C are positive constants independent of time and initial data, but which depend on the other structural parameters of the problem.
Remark 2.6. The proof of (2.7) does not require the validity of the energy identity (2.4), and so it holds also outside the range p ∈ 6 5 , 2 when d = 3, see [19]. Let us now set and endow Y m with the following metric for any ψ 1 , ψ 2 ∈ Y m . Thanks to (2.7) and Theorem 2.2, we can associate with problem (1.6)-(1.9) the solution semiflow where ϕ (t) is the unique weak solution of (1.6)-(1.9 for some positive constants C m and κ, for any ν ∈ (0, 1) and some α ∈ (0, 1) .
(iii) Finite dimensionality: Thus we can immediately deduce the Corollary 2.9. The global attractor A is bounded in V ∩ C α Ω and has finite fractal dimension: To prove Theorem 2.8 we first need to derive a number of properties of the semigroup solution. The first result gives a dissipative estimate in the space L ∞ (Ω) . Moreover, there exists R 0 > 0 (independent of time, τ and initial data) such that S (t) possesses an absorbing ball B L ∞ (Ω) (R 0 ), bounded in L ∞ (Ω).
Proof. Our proof of (2.10) relies on an iterative argument as in [24]. The estimates will be derived assuming sufficiently smooth solutions to (1.6)-(1.9) so that the function |ϕ| p−1 ϕ is also L 2 -summable for each p > 1. The scheme we employ is as follows: let ϕ 0ε ∈ L ∞ (Ω) such that ϕ 0ε → ϕ 0 in H, and such that F (ϕ 0ε ) → F (ϕ 0 ) in L 1 (Ω) as ε → 0. In this case, we can exploit the existence proof of Theorem 2.2 (see [10]) one more time and an a priori L ∞ -estimate from [5, Theorem 2.1] to deduce the existence of a weak solution ϕ ε satisfying (2.2) with the additional essential property Also for practical purposes, C denotes from now on a positive constant that is independent of t, ε, ϕ and initial data, but which only depends on the other structural parameters. Such a constant may vary even from line to line. Further dependencies of this constant on other parameters will be pointed out as needed.
For p > 1, omitting the subscript ε, we multiply equation (1.6) by |ϕ| p−1 ϕ and integrate over Ω, to obtain where C > 0 is independent of p and ε > 0, owing to the assumptions (H1)-(H2) (cf. [ [25,Theorem 2.3]) to derive the following inequality: where t, ξ are two positive constants such that t − ξ/2 k > 0, and C ξ , σ are positive constants independent of k; the constant C ξ is bounded if ξ is bounded away from zero. We can iterate in (2.13) reasoning exactly as in, e.g., [24, Theorem 3.2] (cf., also, [25,Theorem 2.3]). For the sake of completeness, we report a sketch of the argument. Choose any numbers τ ′ > τ > 0 such that ξ = (τ ′ − τ ), t 0 = τ ′ and t k = t k−1 − ξ/2 k , k ≥ 1. Thus, in view of (2.13) we have (2.14) sup Next, define Thus, we can iterate in (2.14) with respect to k ≥ 1 and obtain that Therefore, taking the 2 k -root on both sides of (2.16) and then letting k → +∞ (note that the series in (2.17)-(2.18) are convergent), we deduce for some positive constant C 0 independent of t, k, ϕ, ε, ξ and initial data. In order to prove the first assertion of lemma, we observe that a simple argument [18, Proposition 4, (3.21)-(3.22)] yields, on account of (2.7), that Thus, setting τ ′ = 2τ so that ξ = τ , we readily obtain the first claim (2. , it is not difficult to see that, for any bounded set B ⊂ Y m , there exists a time t * = t * (B) > 0 such that S (t) B ⊂ H, for all t ≥ t * . Next, we can choose τ ′ = τ + 2ξ with τ = t * and ξ = 1, so that C H and C ξ are bounded uniformly with respect to initial data as t ≥ t * . Hence, the L 2 -L ∞ smoothing property (2.19) immediately entails the second assertion of lemma.
We also have Lemma 2.11. Let the assumptions of Theorem 2.2 be satisfied. Then, for every τ > 0, there exists a constant C m,τ,α > 0 such that for some α ∈ (0, 1) . Thus, there exists R 1 > 0 (independent of time, τ and initial data) such that S (t) possesses an absorbing ball Proof. We can rewrite the system (1.6)-(1.8) in the following form Since J ∈ W 1,1 (R d ) and ϕ is bounded by Lemma 2.10, using the fact that a (x) + F ′′ (ϕ) ≥ c 0 , by (H2), it is easy to check that for some positive constants C i which depend only on J W

Moreover, for any bounded set
The following result shows that the semigroup is strongly continuous with respect to the V ′ -metric. Proposition 2.13. Let ϕ i , i = 1, 2, be a pair of weak solutions according to the assumptions of Theorem 2.2. Then the following estimate holds: , for some positive constants κ, C which depend on c 0 and J but are independent of ϕ i (0) .
The crucial step in order to establish the existence of an exponential attractor is the validity of so-called smoothing property for the difference of two solutions (see [37]). In the present case, such a property is a consequence of the following two lemmas. The first result establishes that the semigroup S (t) is some kind of contraction map, up to the term Lemma 2.14. Let the assumptions of Proposition 2.13 hold. Then, for every τ > 0, we have: for all t ≥ 3τ , for some positive constants C, C m,τ , κ which depend on c 0 , Ω and J.
We now need some compactness for the term ϕ 1 − ϕ 2 L 2 ([3τ,t];V ′ ) on the righthand side of (2.30). This is given by Lemma 2.15. Let the assumptions of Proposition 2.13 hold. Then, for every τ > 0, the following estimate holds: Proof. The second term on the left-hand side of (2.33) can be easily controlled by (2.25). Thus we only need to estimate the time derivative. Recall that ϕ satisfies (2.26). Furthermore, in light of Lemmas 2.10 and 2.12, recall that we have Thus, for any test function ψ ∈ D(A N ), using the weak formulation (2.5), there holds This estimate together with (2.25) gives the desired estimate on the time derivative in (2.33).
We now show that the semigroup S (t) is actually uniformly Hölder continuous in the C α -norm with respect to the initial data.
Then, for every τ > 0, the following estimate is valid: for all t ≥ 3τ , where the constants C m,τ , κ and β < 1 are independent of the initial data.
Proof. Using the interpolation [V, V ′ ] 1/2,2 = H, we deduce from estimates (2.25) and (2.34) that , the nonlinearity f = F ′ is controlled by the linear part of the equation (2.26) (no matter how fast it grows) and obtaining the L 2 -L ∞ smoothing property for our dynamical system is actually reduced to the same standard procedure we used in the proof of Lemma 2.10. Indeed, we already have an estimate of the L ∞ -norm of the solution ϕ (t) (due to (2.10)) and, consequently, we do not need to worry about the growth of f = F ′ . In particular, estimate (2.12) also holds for the difference of solutions ϕ = ϕ 1 − ϕ 2 . This observation combined with (2.37) and a proper interpolation inequality between C α Ω ⊂ C α/2 Ω ⊂ L ∞ (Ω) implies the desired inequality (2.36).
The last ingredient we need is the uniform Hölder continuity of t → S(t)ϕ 0 in the C α -norm, namely, where β < 1 and the positive constant C m,τ is independent of initial data, ϕ and t, s.
Proof. According to (2.34), the following bound holds for µ: Consequently, by comparison in (1.6), we have that property of the solutions in [3τ, ∞) and the interpolation inequality We report for the reader's convenience the following abstract result on the existence of exponential attractors [15, Proposition 4.1] which will be used in the following proof and in the other sections as well.
Proposition 2.18. Let H,V,V 1 be Banach spaces such that the embedding V 1 ⊂ V is compact. Let B be a closed bounded subset of H and let S : B → B be a map. Assume also that there exists a uniformly Lipschitz continuous map T : for some L ≥ 0, such that for some γ < 1 2 and K ≥ 0. Then, there exists a (discrete) exponential attractor M d ⊂ B of the semigroup {S(n) := S n , n ∈ Z+} with discrete time in the phase space H.
Proof of Theorem 2.8. In order to apply Proposition 2.18, it is sufficient to verify the existence of an exponential attractor for the restriction of S(t) on some properly chosen semi-invariant absorbing set in Y m . Recall that, by Lemmas 2.10 and 2.12, the ball B 0 := B C a (Ω)∩V (R 0 ) will be absorbing for S (t), provided that R 0 > 0 is sufficiently large. Since we want this ball to be semi-invariant with respect to the semigroup, we push it forward by the semigroup, by defining first the set and then the set On the other hand, due to the results proven in this section, we have for every trajectory ϕ originating from ϕ 0 ∈ B, for some positive constant C m which is independent of the choice of ϕ 0 ∈ B. We can now apply the abstract result above to the map S = S (T ) and H = V ′ m , for a fixed T > 0 such that e −κT < 1 2 , where κ > 0 is the same as in Lemma 2.14. To this end, we introduce the functional spaces and note that V 1 is compactly embedded into V. Finally, we introduce the operator Therefore, due to Proposition 2.18, the semigroup S(n) = S (nT ) generated by the iterations of the operator S : B → B possesses a (discrete) exponential attractor M d in B endowed by the topology of V ′ m . In order to construct the exponential attractor M for the semigroup S(t) with continuous time, we note that, due to Lemma 2.13, this semigroup is Lipschitz continuous with respect to the initial data in the topology of V ′ m . Moreover, by (2.36) and (2.38) the map (t, ϕ 0 ) → S (t) ϕ 0 is also uniformly Hölder continuous on [0, T ] × B, where B is endowed with the metric topology of V ′ m . Hence, the desired exponential attractor M for the continuous semigroup S(t) can be obtained by the standard formula In order to finish the proof of the theorem, we only need to verify that M defined as above will be the exponential attractor for S(t) restricted to B not only with respect to the V ′ m -metric, but also in with respect to a stronger metric. This is an immediate corollary of the following facts: B is bounded in V ∩ C α Ω , the L 2 -C α Ω smoothing property of the map ϕ 0 → S (t) ϕ 0 , and the interpolation inequalities given by (2.40) and for some s = s (ν) ∈ (0, 1). Theorem 2.8 is now proved.
Remark 2.19. The methods used in this section can also be applied to other nonlocal problems which have a variational structure similar to the nonlocal Cahn-Hilliard equation. An interesting case (see [5,Sec. 5] and its references) is a model related to interacting particle systems with Kawasaki dynamics, namely, for some constant β. In fact, in this case the L 2 -L ∞ smoothing property proven in Lemma 2.10 holds again regardless of the value of β. The existence of an absorbing set in V ∩ L ∞ (Ω) for the solution map ϕ 0 ∈ H → ϕ (t) ∈ H of (2.45) can be also established as in [5,Section 5]. Hence, the existence of an exponential attractor for the dynamical system associated with (2.45) can be proven arguing as in Theorem 2.8.

2.3.
Convergence to a single equilibrium. Let ϕ be a weak solution to (1.6)-(1.9) according to Theorem 2.2. In this section we aim to show that that the ω-limit set, is a singleton, where ϕ * is a solution of the stationary problem: (see, e.g., [5,Theorem 4.5]). We employ a generalized version of the Lojasiewicz-Simon theorem proved in [22,Theorem 6] (cf. also [35,Section 4]). The version that applies to our case is formulated in the following.
Lemma 2.20. Let J satisfy (H1) and F ∈ C 2 (R) be a real analytic function satisfying (H2). Then, there exist constants θ ∈ (0, 1 2 ], C > 0, ε > 0 such that the following inequality holds: Proof. We will now apply the abstract result [22,Theorem 6] to the energy functional E (ϕ), which according to (2.3) is the sum of a double-well potential and an interface energy term. In contrast to this feature, we shall split E (ϕ) into the sum of a convex (entropy) functional Φ : H = L 2 (Ω) → R ∪ {∞}, with a suitable effective domain, and a non-local interaction functional Ψ : H → R. To this end, we define the lower-semicontinuous and strongly convex functional with closed effective domain dom(Φ) = L ∞ (Ω) ∩ Y m , and the quadratic functional Ψ : H → R, given by for all ϕ ∈ U and ξ ∈ L ∞ (Ω). The analyticity of DΦ as a mapping on L ∞ (Ω) is standard and can be proved exactly as in, e.g., [16,Theorem 5.1]. Moreover, due to assumption (H2), there holds H , for all ϕ 1 , ϕ 2 ∈ U , and for some positive constant γ. Moreover, computing the second Fréchet derivative is an isomorphism for every ϕ ∈ U . Concerning the (quadratic) function Ψ, we see that We recall that the linear operator ψ → J * ψ is self-adjoint and compact from H to itself and is also compact from L ∞ (Ω) to C 0 (Ω) (cf. Remark 2.1). On the other hand, we also have the following ( is a well defined, bounded from below functional with nonempty, closed, and convex effective domain dom(F ) =dom(Φ) . Unravelling notation in [22,Theorem 6], and observing that the Fréchet derivative from which (2.47) follows.
We can now prove the following convergence result.
Proof. Before we begin the proof, we note that by virtue of Lemma 2.10 and Lemma 2.12, all ϕ * ∈ ω [ϕ] are bounded in L ∞ (Ω) ∩ V . Besides, recalling also the energy identity (2.3), we have E (ϕ (t)) → E ∞ , as t → ∞, and the limit energy E ∞ is the same for every steady-state solution ϕ * ∈ ω [ϕ]. Moreover, we can integrate the energy equality (2.3), By virtue of Lemma 2.20 (cf. also Remark 2.1), we have This, combined with the previous identity, yields for all t > 0, for as long as (2.50) holds. Note that, in general, the quantities θ, C and ε above may depend on ϕ * . Let us set Consequently, using the bound (2.52) and the main equation (1.6), we also obtain In order to finish the proof of the convergence result in (2.48) it suffices to show that it holds in H-norm. Indeed, in this case (2.48) will become an immediate consequence of the L 2 -(L ∞ ∩ V ) smoothing property of the weak solutions and all ϕ * ∈ ω [ϕ] (see Lemmas 2.10 and 2.12). We claim that we can find a sufficiently large time τ > 0 such that (τ, ∞) ⊂ M . To this end, recalling (2.49) and the above bounds, we also have that ∂ t ϕ ∈ L 2 (0, ∞; V ′ ), ∇µ ∈ L 2 (0, ∞; H d ) and, furthermore, for any δ > 0 there exists a time t * = t * (δ) > 0 such that Next, observe that by Lemma 2.12 and Lemma 2.10, there is a time t # > 0 such that Now, let (t 0 , t 2 ) ⊂ M , for some t 2 > t 0 ≥ t * (δ) , |t 0 − t 2 | ≥ 1 such that (2.55) holds (w.l.o.g., we shall assume that t * ≥ t # ). This claim is an immediate consequence of the aforementioned L 2 -(L ∞ ∩ V ) smoothing property and bounds (2.54). Using (2.54) and (2.55), we obtain Therefore we can choose a time t * (δ) = τ < t 0 < t 2 , such that provided that (2.50) holds for all t ∈ (t 0 , t 2 ). Since ϕ * ∈ ω [ϕ], a large (redefined) τ can be chosen such that we have t > τ and ϕ t − ϕ * H ≥ ε if t is finite. On the other hand, in view of (2.57) and (2.58), we have for all t > t ≥ τ , and this leads to a contradiction. Therefore, t = ∞ and by (2.54) the integrability of ∂ t ϕ in L 1 (τ, ∞; V ′ ) follows. Hence, ω [ϕ] = {ϕ * } and (2.48) holds on account of the L 2 -(L ∞ ∩ V ) smoothing property. The proof is finished.
The following existence result holds.
Proof. The argument for α = 0 was given in [19, Theorem 1 and Corollary 1]. In the case α > 0 the proof goes essentially as in [19] with some minor modifications (see Step 1 below). Indeed, the whole idea is to approximate (1.6)-(1.9) by a problem P ǫ which is obtained from (1.6)-(1.9) by replacing the singular potential F with a smooth (regular) potential of polynomial growth F ǫ = F 1ǫ + F 2 , where F 1ǫ is defined in such a way (cf. [19, Lemma 1 and Lemma 2]) that F ′ ǫ → F ′ uniformly on every compact interval included in (−1, 1), and such that the following properties hold: (i) There exist c q , d q > 0, which depend on q but are independent of ǫ, and ǫ 0 > 0 such that (ii) Setting c 0 := α + β + min The approximating problem P ǫ , ǫ > 0, then takes the following form: find a weak solution ϕ ǫ to in Ω.
Recalling (ii) and estimate (3.10), we get for some appropriate constant C J > 0 which is independent of ǫ but depends on ∇J L ∞ (R d ) . Estimate (3.13) yields on account of a suitable Gronwall's inequality, 2 V e −γt + C α,J , ∀t ≥ 0, for some positive constant γ independent of ǫ. This further estimate ensures that we have strong convergence in L 2 (0, T ; H) for some subsequence so that we can identify the nonlinear term in the continuous limit. The existence of a solution to P ǫ is proven in the case of a smooth initial datum.
We can now establish the existence of a solution with an initial datum ϕ 0 ∈ H such that | ϕ 0 | < 1 and F ǫ (ϕ 0 ) ∈ L 1 (Ω) with a sequence ϕ 0j ⊂ V with the same properties and such that (3.15) ϕ 0j → ϕ 0 in H-norm, as j → ∞. Also we take a sequence {J j } ∈ W 1,∞ (R d ) which satisfies (H1) and strongly converges to J in W 1,1 (R d ) as j → ∞. Let {ϕ j,ǫ } be a sequence of solutions associated with {ϕ j,0 }. Arguing as in the proof of Lemma 3.3 below, we can get the estimate for some positive constants C, κ which depend on J j W 1,1 (R d ) and Ω, but are independent of ǫ and i, j. This yields, on account of (3.15), that as j → ∞, we have the strong convergence of the sequence of solutions ϕ j,ǫ for α > 0, to some function ϕ ǫ , i.e., (3.17) ϕ j,ǫ → ϕ ǫ strongly in C ([0, T ]; H) , for every ǫ > 0. Finally, from the preceding estimates (3.10)-(3.12), we can infer that (up to subsequences) as j → ∞. Therefore, arguing as in [10], the convergence properties (3.17)-(3.18) allow us to show that ϕ ǫ is a weak solution to problem P ǫ , with ϕ 0 satisfying the assumptions of Theorem 3.2 and J fulfilling (H1).
Step 2. The passage to limit as ǫ → 0 is actually easier. Indeed, we have already observed that estimates (3.10)-(3.12) and (3.16) hold with constants independent of ǫ > 0. The strong convergence can still be deduced by using the continuous dependence estimate proven here below (see (3.19)). Thus a similar argument works when ǫ goes to zero. We will only mention that in order to pass to the limit in the variational formulation for problem P ǫ , we need to show that |ϕ| < 1 almost everywhere in Q = Ω × (0, T ). This can be done by adapting an argument from [14,Section 4]. We refer the reader to [19,Section 3] for further details.
Uniqueness is an immediate consequence of the following result whose proof goes essentially as in Lemma 2.14.
, for some positive constants C, κ which depend on c 0 and J, but are independent of α ≥ 0.
Proof. We see that ϕ (formally) satisfies the problem: subject to the boundary and initial conditions Arguing as in Lemma 2.14, we obtain, on account of (H10), the following estimate: where κ, C depend on c 0 , Ω and J, but are independent of α ≥ 0. Observe now that we also have F ′ (ϕ) ∈ L ∞ 0, T ; L 1 (Ω) . Therefore we can still deduce (2.29) and the application of Gronwall's inequality to (3.22) entails the desired estimate (3.19) exactly as in Lemma 2.14.
On account of the previous results, we can define a dynamical system on the metric space where m ∈ [0, 1) is fixed and the metric is given by (2.8). Then, for each α ≥ 0 we can also define a semigroup where ϕ (t) is the unique weak solution of (1.6)-(1.9). In fact, arguing as in [19, Section 4, Theorem 2], we deduce the following (−1, 1). Then the dynamical system (Y m , S(t)) possesses a connected global attractor A.

3.2.
Exponential attractors. Note that, according to Theorem 3.4, a global attractor A exists for any α ≥ 0. However, we are able to show its finite dimensionality only in the case α > 0. This assumption is intimately connected with the aforementioned separation property which will allow to handle F ′ on a closed interval of the form [−1 + δ, 1 − δ]. We have the following. (ii) Separation property: there exists δ 0 = δ 0 (m, α) ∈ (0, 1) such that (iii) Exponential attraction: for some positive constants C m and κ, for any s ∈ [2, ∞).
(iv) Finite dimensionality: Remark 3.6. Note that, thanks to the separation property, the assumption that F is bounded on (−1, 1) (see Theorem 3.4) is not needed. First, we derive some (uniform in time) a priori estimates for the weak solutions. For the next result, we also assume that the boundary Γ is of class C 2 (we note that Lemma 3.8 is the only place where this assumption is used; however, see Remark 3.9).
As we mentioned above, we note that (3.25) is actually intended to be satisfied by a standard Galerkin approximation of ϕ ε , in which we should have at least ∂ t ζ ∈ L 2 0, T ; L 2 (Ω) . The required regularity in (3.24) will be then obtained by passing to the limit in the subsequent estimates. Thus, in what follows we shall proceed formally. Testing (3.25) with ψ = N ζ(= A −1 N ζ), then integrating by parts, we obtain 1 2 which yields, thanks to assumptions (H1) and (H10), H , for some positive constant C which depends only on c 0 and J. Thus, using this inequality and exploiting the basic energy identity (3.5), we have Thus, in view of the uniform Gronwall's lemma, we infer From this point on, the constant C will always denote a computable quantity whose expression is allowed to vary on occurrence, depending on the initial data, on α −1 > 0, and on the other fixed parameters of the system. We shall again point out its dependence on various parameters whenever necessary. Therefore, by comparison in (1.6) and on account of (3.29), we deduce that Next, let us test (2.5) by ψ = N (F ′ (ϕ) − F ′ (ϕ) ) to obtain Then note that Therefore, on account of (3.5) and (3.29), the above estimate allows us to infer We can now easily argue as in the proof of Theorem 3.
Remark 3.9. The assumption on Γ ∈ C 2 can be dispensed with so that the result below in Lemma 3.10 also holds for bounded domains with Lipschitz boundary Γ. Indeed, on account of known elliptic regularity theory (cf., e.g., [12]) for problem (1.6), (1.8), we can deduce that µ (t) ∈ L ∞ [τ, ∞); H 1+γ (Ω) , for any γ ∈ 1 2 , 1 . Note that we cannot take γ = 1 without further assumptions on Γ (see [12]). Since Ω ⊂ R d , d ≤ 3, we have H 1+γ (Ω) ⊂ L ∞ (Ω) in the range provided for γ and the argument below in (3.36) still applies. Thus, we can conclude the validity of Lemma 3.10 in the case of a bounded domains with Lipschitz boundary as well.
Step 1. To prove the instantaneous boundedness (3.31), we rewrite equation (1.7) as a first-order ordinary differential equation: Recall that (3.33) is also subject to the initial condition in Ω, and that we have (cf. Theorem 3.2) Next, according to estimate (3.24) and using the embedding H 2 (Ω) ⊂ L ∞ (Ω), we have Step 2. In order to deduce the uniform estimate (3.32) we shall first derive the following dissipative estimate: for all t ≥ 0, for some positive constant C m independent of the initial data, time and α ≥ 0, but which depends on m ∈ (0, 1) such that | ϕ 0 | ≤ m. The proof of (3.38) follows the same lines of [19,Proposition 2] and [10,Corollary 2]. We briefly mention some details. Let us thus multiply equation µ = aϕ − J * ϕ + F ′ (ϕ) + α∂ t ϕ by ϕ in L 2 (Ω) and integrate over Ω. We obtain Observe now that, due to the singular character of F ′ , we can find C F > 0 such that Then, using (3.40), we obtain which is independent of ϕ 0 and time, such that |ϕ (t)| ≤ 1 − δ 1 , a.e. in Ω, for all t ≥ t 0 . Inequality (3.32) is now proven.
In what follows, we derive as in Section 2.2 some basic properties of S (t) which will be useful to establish the existence of an exponential attractor. The following proposition, whose proof goes as in Lemma 2.14, is immediate (see Lemma 3.3).
Proposition 3.11. Let the assumptions of Lemma 3.3 hold. Then we have for all t ≥ 0, where M i := ϕ i (0) , for some positive constants β, C which depend on c 0 and J, but are independent of α.
The following one is also straightforward.
Proposition 3.12. Let the assumptions of Lemma 3.3 be satisfied. Then, for every τ > 0, the following estimate holds: where C m,τ and κ > 0 also depend on c 0 , Ω, α > 0 and J.
Proof. In light of the separation property, the proof goes essentially as the one of Lemma 2.15.
The next lemma gives the uniform Hölder continuity of t → S(t)ϕ 0 with respect to the H-norm. Lemma 3.13. Let the assumptions of Theorem 3.2 be satisfied. Consider ϕ (t) = S (t) ϕ 0 with ϕ 0 ∈ Y m . Then, for every τ > 0, there holds: where the positive constant C m,α,τ is independent of initial data, ϕ and t, s.
Proof of Theorem 3.5. As in Section 2.2, we apply the abstract result of Proposition 2.18. In light of the separation property in Lemma 3.10, it is not difficult to realize that there exists an absorbing set of the following form for a suitable constant δ α . We endow B (δ α , m) with the metric of H = H, and reasoning as above (see Section 2.2), we can suppose that B (δ α , m) is semi-invariant for S (t) for t ≥ 0. On the other hand, due to the results proven in this section, we have sup Remark 3.14. In contrast to the results proved in the case of regular potentials, we cannot show that ϕ(t) is ultimately bounded in V -norm like in the nonviscous case α = 0 (cf. Lemma 2.12 and (3.13)). This can also be understood by formally rewriting the original equation in the following form which shows that this equation is much closer to the nonlocal Allen-Cahn equation (see, for instance, [1,2,4,8,28] and references therein). Moreover, there is a close connection between the viscous nonlocal Cahn-Hilliard equation and the phase-field system investigated in [31], namely, ∂ t (ςϑ + ϕ) = ∆ϑ, (3.51) α∂ t ϕ − J * ϕ + aϕ + F ′ (ϕ) = ϑ, (3.52) in Ω × (0, ∞), where ϑ denotes a rescaled relative temperature and ς > 0. Indeed, if we let ς go to 0 formally, then we get ∂ t ϕ = ∆ϑ.
Thus we obtain that is, the viscous nonlocal Cahn-Hilliard equation. It would be interesting to investigate the connections between the phase-field system (3.51)-(3.52) and equation (3.53) along the lines of what was done for the local equations (see, e.g., [27] and its references).

3.3.
Convergence to a single equilibrium. Also in this case we have all the ingredients to show that each trajectory does converge to a single equilibrium. We can state the following version of the Lojasiewicz-Simon theorem whose proof goes exactly, with some minor modifications, as in Lemma 2.20 (cf. [22] also). . Then, there exist constants θ ∈ (0, 1 2 ], C > 0, ε > 0 such that the following inequality holds: The analog of Theorem 2.21 in the case α > 0 and singular potentials f is where ϕ * is solution to (2.46).
Proof. The proof goes essentially along the lines of Theorem 3.16. Indeed, it is easier since by virtue of (3.54) and the energy identity (3.5), one can establish instead of (2.53) the bound: which entails the integrability of ∂ t ϕ in L 1 (τ, ∞; H). We leave the details to the interested reader (see, also, [16, Section 6]).

Degenerate mobility and logarithmic potential
In this section we consider the model proposed in [23] (see also [29,30]). Thanks to the particular form of the mobility coefficient, the separation property holds even in absence of viscosity (see [35,36]). As a consequence, we can prove the existence of an exponential attractor in this case as well.
The main result of this section is contained in the following Theorem 4.8. Let the assumptions of Theorem 4.2 be satisfied. There exists an exponential attractor M bounded in C α Ω , α ∈ (0, 1) and compact in H, for the dynamical system (Y m1,m2 , S (t)) associated with (4.1)-(4.2), satisfying the following properties: (i) Semi-invariance: S (t) M ⊂ M, for every t ≥ 0.
(iv) Finite dimensionality: Consequently, we also have the following In what follows, we derive as in Sections 2 and 3 some basic properties of S (t) which will be useful in order to establish the existence of an exponential attractor. The following proposition shows that the semigroup S (t) is Lipschitz continuous in the H-norm with respect to the initial data.
Proof of Theorem 4.8. We shall essentially argue as in Section 2.2 by applying Proposition 2.18. We briefly mention the details. In light of the separation property in Theorem 4.4, it is not difficult to realize that there exists a (semi-invariant) absorbing set of the following form B δ := ϕ ∈ Y 0 ∩ C α Ω : δ ≤ ϕ ≤ 1 − δ, a.e. in Ω .
Therefore, it is sufficient to verify the existence of an exponential attractor for S(t) |B δ . Note that due to the above results, we also have sup t≥0 ϕ (t) C α (Ω) + µ (t) L ∞ (Ω) + w (t) W 1,∞ (Ω) ≤ C δ , for every trajectory ϕ originating from ϕ 0 ∈ B δ , for some positive constant C δ which is independent of the choice of ϕ 0 ∈ B δ . We can now apply the abstract result above to the map S = S (T ) and H = H, for a fixed T ≥ T 0 such that e −λ0T < 1 2 , where λ 0 > 0 is the same as in Lemma 4.11. To this end, we introduce the functional spaces and note that V 1 is compactly embedded into V. Finally, we introduce the operator T : B δ → V 1 , by Tϕ 0 := ϕ ∈ V 1 , where ϕ solves (4.1)-(4.2) with ϕ (0) = ϕ 0 ∈ B δ . The maps S, T, the spaces H,V,V 1 thus defined satisfy all the assumptions of Proposition 2.18 on account of Lemma 4.11 (see (4.18)-(4.19)). Therefore, the semigroup S(n) = S (nT ) generated by the iterations of the operator S : B δ → B δ possesses a (discrete) exponential attractor M d in B δ endowed by the topology of H. In order to construct the exponential attractor M for the semigroup S(t) with continuous time, we note that, due to Lemma 4.12 and Proposition 4.10, this semigroup is Lipschitz continuous with respect to the initial data in the topology of H. Besides, the map (t, ϕ 0 ) → S (t) ϕ 0 is Hölder continuous on [0, T ]×B δ , where B δ is endowed with the metric topology of V ′ . Hence, the desired exponential attractor M for the continuous semigroup S(t) can be obtained by the same standard formula in (2.44). Theorem 4.8 is now proved.
In this case, the exponential attraction (iii) and finite dimensionality (iv) of M also holds with respect to the L s ∩ H 1−ν -metric for any s ≥ 2 and ν ∈ (0, 1), on account of (4.10) and (4.25). Let us briefly explain how to get (4.25).
In view of (4.25), this can be improved to a convergence rate in the L s ∩H 1−ν -metric for any s ≥ 2 and ν ∈ (0, 1), i.e., ϕ (t) − ϕ * L p ∩H 1−ν ∼ (1 + t)  The assumption on J in (4.26) can be actually relaxed to also include Newtonian and Bessel-like potentials. In general, the second derivatives of such potentials are not locally integrable, but one may still properly define ∇ 2 J * ϕ as a bounded distribution on L p (Ω) , 1 < p < ∞, using the Calderón-Zygmund theory. In particular, for such distributions there holds ∇ − → V ∈ L p for every 1 < p < ∞ (see, e.g., [3,Lemma 2], [26, Lemma 2.1]), and thus we can also conclude (4.25) for such potentials.
This is a conditional result which requires stronger assumptions on the kernel J and on Ω. For instance, to prove the condition ∂ t ϕ (T 0 ) (H 1 ) * < ∞ one needs to show that ϕ (T 0 ) V < ∞.