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Existence of global attractors and convergence of solutions for the Cahn-Hilliard equation on manifolds with conical singularities

We consider the Cahn-Hilliard equation on manifolds with conical singularities and prove existence of global attractors in higher order Mellin-Sobolev spaces with asymptotics. We also show convergence of solutions in the same spaces to an equilibrium point and provide asymptotic behavior of the equilibrium near the conical tips in terms of the local geometry.


Introduction
In this article, we show existence and regularity of global attractors as well as convergence results for the Cahn-Hilliard equation considered on manifolds with conical singularities. We model such a manifold as a (n + 1)-dimensional compact manifold B with closed boundary ∂B, n ≥ 1, which is endowed with a degenerated Riemannian metric g that, in local coordinates (x, y) ∈ [0, 1) × ∂B on a collar neighborhood of the boundary, has the following expression where [0, 1) ∋ x → h(x) is a smooth family of Riemannian metrics on ∂B. We denote B = (B, g) and ∂B = (∂B, h(0)). The Laplacian on B, in local coordinates (x, y) ∈ [0, 1) × ∂B on the collar part, admits the following degenerate expression where ∆ h(x) is the Laplacian on (∂B, h(x)). The operator ∆ belongs to the class of cone differential operators or Fuchs type operators, see Section 3 for more details. On B we consider the following problem (1.1) u ′ (t) + ∆ 2 u(t) = ∆(u 3 (t) − u(t)), t ∈ (0, T ), which is known as Cahn-Hilliard equation (CH for short); it is a diffusion interface equation that models phase separation of a binary mixture. On classical domains, (1.1) has been generalized and extensively studied in many directions and aspects, such as existence, regularity and convergence of solutions, existence of global attractors, etc. A sufficient number of related results can be found in [17]. However, on singular domains much less is known. Using the theory of cone differential operators, in [21] it was first shown short-time existence of CH on B for the case where h(·) is constant, by employing L p -maximal regularity techniques. Those results were extended to arbitrary B and improved to higher regularity in [22]. Finally, global solutions and smoothing results were proved in [13]. Summarizing those results, let us assume that dim(B) = n + 1 ∈ {2, 3}, choose s ≥ 0 and let the weight γ be as follows where 0 = λ 0 > λ 1 > · · · are the eigenvalues of the boundary Laplacian ∆ h(0) on ∂B. Denote by H η,ρ (B), η, ρ ∈ R, the Mellin-Sobolev space, see Definition 2. Moreover, let R ω and C ω be spaces of smooth functions on B that are locally constant close to the singularities, with values in R and C respectively, see Section 3 for details. Then, for any real-valued u 0 ∈ H s+2,γ+2 (B) ⊕ R ω , there exists a unique global solution in the following sense: for any T > 0 there exists a unique u ∈ H 1 (0, T ; H s,γ (B)) ∩ L 2 (0, T ; D(∆ 2 s )) solving (1.1) on [0, T ] × B. Furthermore, the solution u satisfies the regularity u ∈ Here E ∆ 2 ,γ is an s-independent finite dimensional space consisting of C ∞ (B • )-functions, that in local coordinates (x, y) ∈ [0, 1) × ∂B, take the form ω(x)c(y)x ρ ln k (x), ρ ∈ C, k ∈ {0, 1, 2, 3}, where c ∈ C ∞ (∂B). More precisely, there exists a discrete set of points Z ∆ 2 in C, determined only by the family of metrics h(·), such that the exponents ρ coincide with the set Z ∆ 2 ∩{z ∈ C : Re(z) ∈ [ n−7 2 −γ, n−3 2 −γ)}. The exponents k are also determined by h(·). In particular, when h(·) = h is constant, the set Z ∆ 2 and the exponents k associated to each ρ ∈ Z ∆ 2 , are determined by n and the spectrum of Consequently, both spaces D(∆ 2 s ) and E ∆ 2 ,γ are determined explicitly by h(·) and γ, see Corollary 21 for details.
These results allow us to define for any s ≥ 0 a semiflow T : , see e.g. [32, Chapter 1, Section 1.1] for more details on semiflows, also known as semigroups. Let X s 1,0 be the space of all realvalued functions u ∈ H s+2,γ+2 (B) ⊕ R ω such that´B udµ g = 0, where dµ g is the measure associated with the metric g. Then T can be restricted to X s 1,0 , see Section 4. Our main results are the following. Theorem 1. Let s ≥ 0, γ be as (1.2) and D(∆ 2 s ) be the bi-Laplacian domain described in (1.5)-(1.9). (i) (Global attractor) The semiflow T : [0, ∞) × X s 1,0 → X s 1,0 has an s-independent global attractor A ⊂ ∩ r>0 D(∆ 2 r ). Moreover, if B is a bounded set of X s 1,0 , then for any r > 0, T (t)B is, for sufficiently large t, a bounded set of D(∆ 2 r ) and lim The definition of global attractor is recalled in Section 4. For proving part (i) of Theorem 1.1, we follow the strategy of Temam [32] to obtain estimates in a lower regularity space H −1 0 (B), see Definition 9, and of [31] for obtaining higher regularity. For convergence to equilibrium, we first obtain the Lojasiewicz-Simon inequality due to [30], and proceed as [6], [16] and [25].
Though the strategies are mostly well established, the technical results that allow us to use them in the context of conical singularities are not, and, therefore, the strategies have to be adapted to this situation. For this reason new results on interpolation and embedding of Mellin-Sobolev spaces are developed in this article.
In Section 2, we define suitable function spaces to work on conic manifolds and study their interpolation and embedding. Section 3 is devoted to the domain description and the properties of the Laplacian and bi-Laplacian. Part (i) of Theorem 1 is proved in Section 4 and part (ii) in Section 5.

Function spaces
Fix a smooth non-negative function ω ∈ C ∞ (B) supported on the collar neighborhood (x, y) ∈ [0, 1) × ∂B such that ω depends only on x and ω = 1 near {0} × ∂B. Moreover denote by C ∞ c the space of smooth compactly supported functions and by H s , s ∈ R, the usual Bessel potential spaces defined using the L 2 -norm.
Definition 2 (Mellin-Sobolev spaces). Let γ ∈ R and consider the map is defined and finite, where 2B is the double of B and * refers to the push-forward of distributions. Different choices of ω, covering and partition of unity give us the same spaces with equivalent norms. The space H s,γ (B) is a Banach algebra, up to an equivalent norm, whenever s > (n + 1)/2 and γ ≥ (n + 1)/2, see [23, Lemma 3.2]. If s ∈ N, then H s,γ (B) coincides with the space of all functions u in H s loc (B • ) that satisfy In Section 3, we will associate the Mellin-Sobolev spaces with the Laplacian and bi-Laplacian.
We define the spaces L p (B) using the measure dµ g induced by the metric g. Note that dµ g = det(h(x))x n dxdy on the collar neighborhood. Finally, recall that the inner product in Besides the Mellin-Sobolev spaces, we define the following space.
where ∇ and ·, · are defined by the conical metric g.
We investigate now certain properties of the space H 1 (B) and its connection with H s,γ (B).
Remark 5. For the following computations, we note that (1) The boundedness of´B |u| 2 dµ g is equivalent to (2) If´B |u| 2 dµ g < ∞, then the boundedness of´B ∇u, ∇u dµ g is equivalent to The last statement can be easily proved once we recall that in local coordinates of [0, 1) × ∂B we have Along this paper, we use ֒→ and c ֒→ to denote continuous and compact embedding, respectively. We recall that H s,γ (B) ֒→ H s ′ ,γ ′ (B), when s ≥ s ′ and γ ≥ γ ′ , and H s,γ (B) c ֒→ H s ′ ,γ ′ (B), when s > s ′ and γ > γ ′ , see [28, Theorem 2.1.53].
Proposition 6. For any β < 1, the following inclusions hold In particular, Proof. We proceed in several steps. Let us denote by C positive constants that can change along the proof. For simplicity we ignore the term det(h(x)) in the proof, as it is uniformly bounded from above and below, and we abuse the notation´∂ B since the computations are made in local coordinates. We also note that it suffices to check the inclusion on the collar neighborhood and for functions u ∈ C ∞ c (B • ).
Step 1: Hence it is clear thatˆB which implies thatˆB We conclude that u H 1 (B) ≤ C u H 1,1 (B) .
Step 2 : For each ε > 0, we have x ε ∈ H 1 (B). Let 0 < r < 1 and χ r : B • → [0, 1] be such that χ r (x, y) = 1 − ω(x/r), for (x, y) ∈ [0, 1) × ∂B and χ r be equal to 1 outside the collar neighborhood. It is enough to prove that lim r→0 χ r x ε = x ε in H 1 (B), as χ r x ε ∈ C ∞ c (B • ). For this, we must prove that Note that (i) follows directly from the dominated convergence theorem and (ii) is identically zero. For (iii), we have that the integral is smaller or equal to two timeŝ Only the last term is important, as we can handle the first one directly with dominated convergence theorem. Note that |ω(x)x ε ∂ x χ r | 2 x n = |ω(x)x/r(∂ x ω)(x/r)| 2 x 2ε+n−2 and that, for n ≥ 1, the integrand is smaller than the integrable function x∂ The result now follows again by the dominated convergence theorem.
Step 3 : C ω ֒→ H 1 (B). It suffices to show that the constant function equal to one belongs to Again (i) follows directly from the dominated convergence theorem, (ii) is identically zero as the functions do not depend on y, and the integral in (iii) is smaller than two timeŝ The first term can be dealt again by dominated convergence. For the second one, note that and the last term goes to zero, as ε goes to zero.
Step 4 : If β < 1, then H 1 (B) ֒→ H 1,β (B). By density, it is enough to show that there is a constant If k + |α| = 0, then as The last integral is finite for n ≥ 1 and β < 1.
In particular, Lemma 7 (Poincaré-Wirtinger inequality). There is a constant C > 0 such that The proof follows the same argument as in the proof of [8, Theorem 1 of Section 5.8], using the fact that H 1 (B) is compactly embedded in H 0,0 (B).
where C is identified with the set of constant functions. Moreover, applying Lemma 7 with (u) B = 0, we see that the map Using the fact that In order to prove that T u is continuous, let us assume, without loss of generality, that −1 < β ≤ 0. We denote by I −β : H 1 (B) → H 0,−β (B) the continuous inclusion from Proposition 6. Then, we have that Therefore T u is continuous as it is the composition of continuous functions. The continuity of T u follows similarly. The fact that . A version of Gauss theorem can also be proved for H 1 (B). It simplifies and improves [13,Lemma 4.3].
Theorem 11 (Gauss theorem). Let u and v belong to H 1 (B) and ∆v ∈ H 0,γ (B), for some γ > −1. ThenˆB Proof. Without loss of generality, we assume that −1 < γ ≤ 0. First we note that for v and u in and, therefore, also in D ′ (B • ). Hence the equality between the first and the third term of (2.5) still holds , the sequence will also converge to u in H 0,−γ (B). Using the duality of H 0,−γ (B) and H 0,γ (B), (2.5) and (2.6), we obtain our result.
Then, the operator ∆ can be easily identified with −R.
Corollary 13. There is a constant C > 0 such that Proof. This follows from the continuous inclusion H 1 0 (B) ֒→ H −1 0 (B), provided by Propositions 6 and 10, and Proposition 12.
Proof. First, we note that . Therefore, it is enough to understand the behavior of elements of H 1,γ (B) in the neighborhood of the conical tip. We have where we recall that {φ j } j∈J is a partition of unity of ∂B and ω is as Definition 2. A change of variables which shows the claim.
Corollary 15. The following continuous inclusions hold: Proof. We have seen that

2.2.
Interpolation for sums of Banach spaces. In this section, we show an abstract result on interpolation of sums of Banach spaces. Later on we will apply it to provide a complete description of the domains of the fractional powers of the Laplacian.
Let X and Y be Banach spaces continuously embedded in a Hausdorff topological vector space V . Denote by F (X, Y ) the set of all functions f : {z ∈ C : Re(z) ∈ [0, 1]} → X + Y that are continuous, bounded and whose restriction f : {z ∈ C : Re(z) ∈ (0, 1)} → X + Y is holomorphic. In addiction, if f ∈ F (X, Y ), then we assume that the functions t ∈ R → f (it) ∈ X, t ∈ R → f (1 + it) ∈ Y are continuous and bounded. By the Hadamard three-lines theorem, any f ∈ F (X, Y ) that satisfies either f (it) = 0, ∀t ∈ R, or f (1 + it) = 0, ∀t ∈ R, must be identically zero.
Lemma 16. Let V be a Hausdorff topological vector space and X, Y , F be Banach spaces continuously [14,Chapter 2] for definition and basic properties of complex interpolation. For the other direction, let x ∈ [X, Y ⊕ F ] θ . Then x = f (θ) for some f ∈ F (X, Y ⊕ F ). We remark that X + (Y ⊕ F ) = X ⊕ F , also topologically. Let P : X ⊕ F → F be the continuous projection on F . The projection when restricted to Y ⊕ F is also continuous.
Theorem 17. Let Y ֒→ X be Banach spaces. Suppose that X = Z ⊕ F , where Z and F are closed subspaces of X, and Concerning the other direction, let Note that the maps I − P : By the definition of complex interpolation, we have In (1) we have used the Lemma 16 and in (2) the embedding Z ֒→ X.
On the other hand, Remark 18. For the rest of the paper, we will use the above theorem when F ֒→ X is a finite dimensional space. In this case, it is well known that there exists a closed subspace Z of X such that X = Z ⊕ F . For a real interpolation analogue of Theorem 17, we refer to [24, Lemma 7.2].

Realizations of the Laplacian and bi-Laplacian
We start with some basic concepts on analytic semigroup theory. Next we recall some basic facts from the cone calculus, for more details we refer to [10,11,12,15,26,27,29]. The Laplacian ∆, as a cone differential operator, acts naturally on scales of Mellin-Sobolev spaces. Let us consider it as an unbounded operator in H s,γ (B), s, γ ∈ R, with domain C ∞ c (B • ). Denote by ∆ s,min the minimal extension (i.e. the closure) of ∆ and by ∆ s,max the maximal extension, defined as usual by D(∆ s,max ) = u ∈ H s,γ (B) | ∆u ∈ H s,γ (B) .
An important result in the field of cone differential operators tells us that those two domains differ in general, unlike the case of closed manifolds. In particular, there exists an s-independent finitedimensional space E ∆,γ , that is called asymptotics space, such that D(∆ s,max ) = D(∆ s,min ) ⊕ E ∆,γ .
More precisely, E ∆,γ consists of linear combinations of smooth functions in B • . Those functions vanish outside the collar neighborhood and, in local coordinates (x, y) ∈ [0, 1) × ∂B, can be written as ω(x)c(y)x −ρ log k (x). The function ω is the cut-off function defined on Section 2, c ∈ C ∞ (∂B), ρ ∈ {z ∈ C | Re(z) ∈ [ n−3 2 − γ, n+1 2 − γ)} and k ∈ {0, 1}. Here, the metric h(·) determines explicitly the exponents ρ. As for the minimal domain, it can be proved that D(∆ s,min ) = H s+2,γ+2 (B), whenever A suitable choice of the domain of the Laplacian is given below. For this, we denote by C ω the finite dimensional space of functions that are equal to zero outside the collar neighborhood and that close to the singularities are expressed by N j=1 c j ω j , where N is the number of connected components of ∂B and ω j are the restrictions of ω to these components. Also denote by R ω the subspace of C ω such that c j ∈ R.  where in (1) we have used reiteration and the BIP property of −A s and in (2) we have used Theorem 17. We also denote X ∞ 1 := ∩ s≥0 X s 1 and X ∞ 2 := ∩ s≥0 X s 2 .

Existence and regularity of the global attractors
For the rest of the paper γ is fixed and satisfies (1.2). The constants C > 0 may change along the computations.
In this section we prove part (i) of Theorem 1. In the sequel all the spaces we use are the real parts of the ones defined previously. Recall that a global attractor for a semiflow T : [0, ∞)× X → X defined on a Hilbert X is a compact set A ⊂ X such that T (t)A := {T (t)x : x ∈ A} = A, for all t ≥ 0, which, moreover, attracts all bounded sets B ⊂ X in the following sense: If it exists, then it is unique.
As mentioned in the introduction, for any s ≥ 0, we can define a semiflow T in X s 1 . It is convenient, however, to restrict T to a smaller space. For this reason, we first prove the following proposition.  Consider the Hilbert space . By Proposition 22, T (t)X s 1,0 ⊂ X s 1,0 . Therefore, T restricts to a semiflow on X s 1,0 . Concerning the existence of global attractors, we recall the following result.
For two Hilbert spaces X and Y such that Y ֒→ X and a semiflow T : [0, ∞) × X → X, we define the ω-limit set ω Z (B) of B ⊂ X, where Z = X or Y , by If Z = Y , then the above definition requires that T (t n )x n ∈ Y for all n ∈ N. In order to show existence and regularity of global attractors, we prove the following variation of [20,Theorem 10.5]. Proof. First we show that ω X (K) = ω Y (K). Since Y ֒→ X, the definition given by (4.1) implies that On the other hand, if x ∈ ω X (K) and lim n→∞ T (t n )x n − x X = 0 for some t n → ∞ and {x n } n∈N ⊂ K, then T (t n )x n ∈ K for all t n > t K . By the compactness of K in Y , some subsequence Let A := ω X (K) = ω Y (K). Using the notation C Y for the closure of C in Y , (4.1) says that which implies that A is a non-empty compact set of Y -and also of X -since it is the intersection of decreasing non-empty compact sets. The invariance of A follows easily from (4.1), see also [9, Proposition 1.1.1]. Finally, the fact that A attracts bounded sets of X in Y -and in X as well -follows from the arguments of [20,Theorem 10.5], which we provide for completeness. Let us suppose that lim t→∞ sup b∈B inf a∈A T (t)b − a Y = 0 does not hold for some bounded set B ⊂ X. Then, we can find sequences t n → ∞, {b n } n∈N ⊂ B and ε 0 > 0 such that inf a∈A T (t n )b n − a Y > ε 0 for all n. But for t n > t B and b ∈ B, then T (t n )b ∈ K which is a compact set in Y . Hence a subsequence {T (t nj )b nj } j∈N converges in Y . As T (t nj )b nj = T (t nj − t n )T (t n )b nj and T (t n )b nj ∈ K, we conclude that the subsequence {T (t nj )b nj } converges to an element of ω Y (K) = A, which gives us a contradiction. Connectness follows exactly as in [20].
For our purposes we use the following consequence.
(ii) There are Hilbert spaces Y and W such that Y c ֒→ W ֒→ X α and a constant C Y > 0 with the following property: for every R > 0, there exists t R > 0 such that if u 0 ∈ X α and u 0 Xα ≤ R, then u(t, u 0 ) ∈ Y and u(t, u 0 ) Y ≤ C Y , for all t > t R .
Then the semiflow T : [0, ∞) × X α → X α defined by T (t)u 0 = u(t) has a global connected attractor A that is contained in W and attracts bounded sets of X α in W .
Proof. We have to show that the conditions of Theorem 23 are satisfied. As Y c ֒→ W , the bounded set K 0 := {x ∈ Y : x Y ≤ C Y } is a relative compact subset of W . We define K to be the closure of K 0 in W . Let B ⊂ X α be a bounded set, i.e. there exists R > 0 such that u 0 Xα ≤ R, for all u 0 ∈ B. By our assumptions, there exists The above corollary will now be applied to the proof of the following theorem, from which part (i) of Theorem 1 will follow. Below in this section 0 < α 0 < 1 will be always as in Corollary 21.

Proposition 26.
There is a constant κ with the following property: for every In order to proceed to the proof of Theorem 25, we write (1.1) as where A s : X s 2 → X s 0 is given by A s = −(1−∆ s ) 2 and F : X s 1 → X s 0 is given by F (u) = ∆ s (u 3 −3u)+u. It is well know, see [18,Theorem 6.13], that for some δ > 0 depending on A s , the fractional powers satisfy Lemma 27. Let 0 ≤ σ ≤ α ≤ β < 1, 0 <t < t, δ > 0 be as in (4.4) and u ∈ C([t, t], D(A s )). Then for some constants C σ , C α,β only depending on α, β, σ and s.
Proof. The proof follows by induction in s. Assume that for some s ≥ 0, there is a constant κ s > 0 with the following property: for every R > 0, there existst R,s > 0 such that if u 0 ∈ X 0 1,0 and u 0 H −1 0 (B) ≤ R, then u(t, u 0 ) H s+2,γ+2 (B)⊕Cω ≤ κ s , for all t >t R,s . We have seen that this is true for s = 0 by Proposition 28.
Proof. (of part (i) of Theorem 1). We check the conditions of Corollary 24 for (4.3). Here we use α = 1/2, so that X α = X s 1 , and X α = X s . For the s-independence, let s 1 > s 2 ≥ 0. As X s1 1 ֒→ X s2 1 is continuous and A s1 is compact in X s1 1,0 , we conclude that A s1 is also compact in X s2 1,0 . Consider now a bounded set B ⊂ X s2 1,0 . Due to Theorem 25, there existst > 0 such that the set T (t)B is a bounded set of X s1 1,0 . Therefore, for t ≥t, we have sup b∈B inf a∈A s 1 Finally, as T (t)A s1 = A s1 , we conclude that A s1 is a global attractor for the semiflow T in X s2 1,0 . By uniqueness of global attractors A s1 = A s2 .

Convergence to the equilibrium
In this section, we prove part (ii) of Theorem 1. We first state an abstract result from [16]. Let V and H be real Hilbert spaces such that V is densely and continuously embedded to H. We recall that an element x ∈ H defines a continuous linear functional in V by y ∈ V → (y, x) H ∈ R. Under this, we have V The above inequality is called the Lojasiewicz-Simon inequality at 0. In our application, the function E of Theorem 30 will be related to the Lyapunov (energy) functional defined for the Cahn-Hilliard equation. In this section, we always assume that dim(B) ∈ {2, 3} and work with the subspaces of real functions. In order to apply the Theorem 30, we need the following technical lemma. Proof. Let β > 0 and T u : In (1) we have used Hölder inequality and in (2) Corollary 15. Therefore we have As H 1 0 (B) ֒→ L 4 (B), we can define the Lyapunov function L : H 1 0 (B) → R by (4.2). It is a real analytic function, as it is the composition of linear and multilinear functions. The following theorem is our main result of this section. Given u 0 ∈ X 0 1,0 , we denote by ω(u 0 ) the ω-limit set ω X 0 1,0 ({u 0 }). Theorem 32. Let u 0 ∈ X 0 1,0 . If ϕ ∈ ω(u 0 ), then there exist constants c, σ > 0 and θ ∈ (0, 1/2] such that the following inequality holds: The argument is standard and can be found e.g. in [6] and [25]. We only stress here the necessary changes for the conical singularities situation.
We The proof of the above expressions uses Theorem 11, the Mellin-Sobolev embeddings from Corollary 15 and the identification of Proposition 10.
Having the Lojasiewicz-Simon inequality, we can prove the convergence theorem below.
Proof. The proof follows from the arguments in [6, Section 3]. We sketch here the steps for the convenience of the reader. Let L ∞ = lim t→∞ L(u(t)). Then L ∞ ≤ L(u(t)) for all t ∈ [0, ∞). We define the function H : [0, ∞) → R by where θ ∈ (0, 1/2] as in Corollary 33. The function H is non-negative and non-increasing, as