ON THE STRONG-TO-STRONG INTERACTION CASE FOR DOUBLY NONLOCAL CAHN-HILLIARD EQUATIONS

. We consider a doubly nonlocal Cahn-Hilliard equation for the nonlocal phase-separation of a two-component material in a bounded domain in the case when mass transport exhibits non-Fickian behavior. Such equations are important for phase-segregation phenomena that exhibit non-standard (anoma- lous) behaviors. Recently, four diﬀerent cases were proposed to handle this important equation and the two levels of nonlocality and interaction that are present in the equation. The so-called strong-to-weak interaction case (when one kernel is integrable in some sense while the other is not) was investigated recently for the doubly nonlocal parabolic equation with a regular polynomial potential. In this contribution, we address the so-called strong-to-strong in- teraction case when both kernels are strongly singular and non-integrable in a suitable sense. We establish well-posedness results along with some regularity and long-time results in terms of ﬁnite dimensional global attractors.


1.
Introduction. Phase-separation in binary materials is considered a central problem in materials science. In this, the Cahn-Hilliard equation (CHE) finds itself center-stage in a three-act "play". The first act starts in the late 1950's with an elegant phenomenological derivation of the equation by Cahn and Hilliard [3]. Since then it has found applications in other important physical phenomena, in image processing, fluid dynamics, population dynamics, and the list goes on. The basic form of CHE is very well-known to the scientific community; it reads as a parabolic equation that is fourth-order in space and the corresponding boundary value problem in a bounded domain requires an addition of two (usually no-flux) boundary conditions. Rather than giving a full account of the enormous scientific literature, we refer the reader to [4,19] where a complete description of the most up-to-date analytical and numerical studies has been undertaken in detail for the standard Cahn-Hilliard equation (CHE). The second act is the most important and brings us closer to the mid 1990's when Giacomin and Lebowitz [11] gave a rigorous physical derivation of the Cahn-Hilliard equation, by starting from a microscopic model for a lattice gas with long-range Kac smooth potentials J ∈ C 2 (R n ) , J (x) = J (−x). This is the nonlocal Cahn-Hilliard equation; it is more general than the classical CHE proposed earlier by Cahn and Hilliard [3], in particular, since at least formally the CHE is a "local first-order" approximation of the nonlocal CHE in a suitable ∂ t ϕ = −A (µ) , µ = B (ϕ) + F (ϕ) , in X × (0, ∞) . (1.5) Here, the nonlocal operator A is defined as which needs to be understood in a principal value sense if K (x) = K (−x) and K is non-smooth, say if K / ∈ L 1 (X). In Gal [8], (1.5) is dubbed as the doubly nonlocal Cahn-Hilliard equation; it was also observed that (1.5) occurs as a generalization of the nonlocal CHE (1.1). A further classification of the physical relevant cases with respect to the heterogeneous pair (K, J) for (1.5) was also given in [8]. In the case when X ⊂ R n is a bounded domain with non-empty Lipschitz continuous boundary ∂X, we recall them as follows: (1) The strong-to-weak interaction case: K / ∈ L 1 loc (R n ) and J ∈ L 1 loc (R n ) ; (2) The strong-to-strong interaction case: both K, J / ∈ L 1 loc (R n ) ; (3) The weak-to-strong interaction case: K ∈ L 1 loc (R n ) and J / ∈ L 1 loc (R n ) ; (4) The weak-to-weak interaction case: both K, J ∈ L 1 loc (R n ) . Furthermore, the strong-to-weak interaction case (1) for (1.5) was settled completely in [8] when X ⊂ R n is a bounded domain with Lipschitz continuous (nonempty) boundary ∂X and both K, J are symmetric. In particular, we gave a unified analysis to establish sharp results in terms of existence, regularity and stability (with respect to the initial data) of properly-defined solutions. There we have also discussed and derived sufficient conditions for problem (1.5) to possess finite dimensional global and exponential attractors, and for solutions to eventually convergence to single steady states as time goes to infinity. In this contribution, we aim to continue that investigation by addressing the next difficult case (2). To treat this case rigorously and to develop a complete theory for well-posedness and long-term behavior of solutions we will require new methods to solve it. In fact, in our attempt to solve case (2) we have stumbled upon further generalizations of (1.5) that include some cases that were not covered before in the scientific literature. This will become evident to the reader in the subsequent sections (see also Tables 1,2). Our main goal is then to develop well-posedness and long-time dynamical results for the equation (1.5) associated with a general class of self-adjoint (nonnegative) operators A, B, and then subsequently recover results for the case (2) of interest as a particular case. In doing so, we shall assume that (X, m) is also a σ-finite measure space. Furthermore, using specific, easy to check, assumptions about the pair (X, m) , and the local and/or nonlocal operators A, B, we will be then introduced to specific models, for which our results are sharp (cf. Subsection 2.2). We emphasize that our assumptions on A, B and/or (X, m) we employ are of general character, and as a result do not require a specific form as suggested by the examples in Tables 1, 2 (cf. Section 3); this abstraction allows to recover the doubly nonlocal Cahn-Hilliard equation (1.5), the classical Cahn-Hilliard equation and the nonlocal Cahn-Hilliard equation (1.1) among all of the existing phase-transition models, as well as to represent a much larger family of nonlocal models, of the form (1.5), that have not been explicitly studied anywhere in detail. For clarity, we list in Tables 1, 2 the corresponding operators and the special cases covered by our theory. To this end, let us briefly define the operators of interest below. Other examples of operators A, B are provided in Section 2 and include also local and nonlocal operators defined over compact manifolds X without boundary. We first denote by −∆ X,N the self-adjoint nonnegative operator that is the realization of the Neumann Laplacian −∆ X with the no-flux boundary condition (1.4), cf. Subsection 2.2. To briefly define the so-called regional fractional Laplacian operator, let us fix s ∈ (1/2, 1) , and set L(X) := {u : X → R measurable, X |u(x)| (1 + |x|) n+2s m (dx) < ∞}.
For u ∈ L(X), we define the regional fractional Laplacian (−∆) s X by the formula (−∆) s X u(x) = P.V. C n,s X (u (x) − u (y)) |x − y| n+2s m (dy) , x ∈ X, (1.7) with a given normalized constant C n,s (see Subsection 2.2). Associated with formula (1.7) and a smooth function u ∈ C 1 (X), one may further define the so-called fractional normal derivative N 2−2s u (cf. [14]). It turns out that a Green type formula involving the fractional normal derivative holds for the regional fractional Laplace operator (−∆) s X [14], see Subsection 2.2. This can be used to define a self-adjoint nonnegative operator (−∆) s X,N , that is the realization of (−∆) s X with the fractional Neumann-type boundary condition N 2−2s u = 0 on ∂X (see [22]; cf. also [12]). To the best of our knowledge, the second model in Table 2 was also investigated in [1] in the case when F belongs to the class (i) defined earlier. Nevertheless it is worthwhile to mention that we generalize even results for the classical Cahn-Hilliard equation (see Table 1) which generally assumed a scenario in which X is quite smooth. This is no longer required in our framework, and besides, it also allows to treat important cases of phase segregation phenomena beyond the ones usually found in the scientific literature. In particular, all of our results on well-posedness, regularity and long-time behavior in terms of finite dimensional global attractors apply to these models in Table 1 and Table 2, respectively. Otherwise, our assumptions on the operators A, B (according to Section 2.2) allow us to deduce comparable results for other ("local-nonlocal" or "doubly nonlocal") phase segregation models that have never been considered before. Table 1. X ⊂ R n is a bounded domain with Lipschitz continuous boundary ∂X. The general model covered is the mass-conserved one given by (1.5) with the following choices of operators A, B. A physically relevant choice that satisfies our assumptions is the double-well potential F (s) = θs 4 − θ c s 2 , 0 < θ < θ c . The remainder of the paper is structured as follows. In Section 2 we define the functional framework needed for our approach, explain its basic properties (see Subsection 2.1) and provide many examples of nonlocal operators (see Subsection 2.2). Section 3 is dedicated to the regularity of weak solutions as well as the existence of strong solutions (Subsection 3.1) while Subsection 3.2 is devoted to the long-term behavior of weak solutions for (1.5) in terms of finite dimensional attractors. In the final Subsection 3.3, we briefly address the problem in the case when F is a singular logarithmic potential.
2. Functional framework. Subsection 2.1 of this section contains a comprehensive account of the theory of (symmetric) Dirichlet forms and then further develops some general (useful) properties that will be needed in the subsequent sections. In Subsection 2.2, actual examples will be presented to emphasize the importance of this analytic theory in the general Cahn-Hilliard theory of nonlocal phase-transition.
2.1. Dirichlet forms and Markovian semigroups. We introduce the notion of Dirichlet form on an L 2 -type space (see [6,Chapter 1]). To this end, let X be a locally compact metric space and m a Radon measure on X such that supp(m) = X. Let L 2 (X, m) be the real Hilbert space with inner product (·, ·) and let E A with domain D(E A ) =: V A be a bilinear form on L 2 (X) = L 2 (X, m).
Definition 2.1. The form E A is said to be a Dirichlet form if the following conditions hold: Remark 2.2. We make the following important remarks.
• Clearly, D (E A ) = V A is a real Hilbert space with inner product E A,λ (u, u) for each λ > 0. We recall that a form E A which satisfies (a)-(c) is closed and symmetric. If E A also satisfies (d), then it is said to be a Markovian form. • When E A is closed, (d) is equivalent to the following more simple condition: call v ∈ L 2 (X) a normal contraction of u ∈ L 2 (X) if some Borel version of v is a normal contraction of some Borel version of u ∈ L 2 (X), that is, |v (x)| ≤ |u (x)| , for all x ∈ X, and It is well-known that there is a one-to-one correspondence between the family of closed symmetric forms E A on L 2 (X) and the family of non-negative (definite) self-adjoint operators A on L 2 (X) in the following sense: From now on, we shall refer to (E A , V A ) as a Dirichlet space whenever E A is a Dirichlet form on D (E A ) = V A in the sense of Definition 2.1. We introduce some further terminology. Let Y, Z be two Banach spaces endowed with norms · Y and · Z , respectively. We denote by Y −→ Z if Y ⊆ Z and there exists a constant C > 0 such that u Z ≤ C u Y , for u ∈ Y ⊆ Z. In particular, this means that the injection of Y into Z is continuous. In addition, if the injection is also compact we shall denote it by Y c −→ Z. By the dual Y * of Y , we think of Y * as the set of all (continuous) linear functionals on Y . When equipped with the operator norm · Y * , Y * is also a Banach space.
The statement is then a simple consequence of Ehrling's lemma.
At this point it is also useful to recall the following result (see [17]). Lemma 2.4. Let Ξ : [k 0 , ∞) → R be a nonnegative, non-increasing function such that there are positive constants c, υ > 0 and δ > 1 such that The following result is also basic but we choose to give a proof for the sake of completeness. However, we note that the conclusion of this statement can be also checked directly in specific cases.
Proof. To prove the first claim, i.e., u k ∈ V A it suffices to show that both |u| , u + = max {u, 0} ∈ V A for as long as u ∈ V A . Indeed, since u ∈ V A and |u| is a normal contraction of u, it follows by Definition 2.1, property (d) , that |u| ∈ V A . In this case, |u| is also a normal contraction of u + and so by the same property, u + ∈ V A . We now need to show the second claim (2.2). Define two sets A k := {x ∈ X : |u (x)| > k} and B k := X\A k and observe that u k = (|u| − k) sgn (u) on A k , while u k = 0 on B k , for each k ≥ 0. We further split We easily see that Our next goal is to investigate a nonlinear elliptic problem associated with the Dirichlet form E A as follows: where h ∈ L s (X, m) for some s > 1. Here, f ∈ C 1 (R) is a nonlinear function which satisfies suitable assumptions (see below).
Definition 2.6. We say that u is a bounded generalized solution of (2 Theorem 2.7. Let the assumptions of Proposition 2.5 be satisfied and suppose that the Dirichlet form E A is also coercive in the following sense:

4)
for some β 0 > 0, λ ∈ [0, α 0 ] with α 0 > 0 as in (2.5). Assume also the following conditions: Then problem (2.3) has at least one bounded solution in the sense of Definition 2.6 provided that Moreover, the following estimate holds: for some constant C > 0 independent of u and h.
Corollary 2.8. Under the assumptions of Theorem 2.7, there exists a strong solution of (2 for some function Q > 0 independent of u and h.
Proof of Theorem 2.7. We first approximate the nonlinear function f by a globally Lipschitz function f ε , ε > 0, given by f for t > ε −1 . Clearly, f ε ∈ C 1 (R) and f ε (t) ≤ C ε , for all t ∈ R, with a constant C ε > 0 that may generally explode as ε → 0 + . On the other hand, assumption (2.5) yields, for ε ∈ (0, ε 0 ] and for some sufficiently small ε 0 = ε 0 (t 0 , α 0 ), that for all |t| ≥ t 0 > 0. Next, we replace f by f ε into (2.3) and consider the corresponding approximate problem Au ε + f ε (u ε ) = h in X, for which we first establish the existence of at least one solution u ε ∈ V A . In a second step, we shall derive bounds for the solution u ε that are uniform in ε > 0 so that we can pass to the limit as ε → 0 to arrive at the final conclusion of the theorem. For practical purposes, in this proof C > 0 denotes a positive constant that is independent of ε. Such a constant may vary even from line to line and its further dependence on other parameters shall be pointed out as needed.
In order to prove the existence of at least one solution u = u ε to Au is also continuous on V A . Therefore, there exists an operator A λ : where ·, · denotes the duality between V A and V * A . Moreover, A λ maps bounded subsets of V A into bounded subsets of V * A and A λ is coercive by (2.4): Thus, by a classical result due to Lions and Leray (see, e.g., [23,Chapter 27]) together with the fact that the embedding V A → L 2 (X) is compact, we deduce that the operator A λ is also surjective; in particular, A λ : V A → V * A is (pseudo)monotone, coercive, continuous and bounded, and for every h is also strongly continuous owing to the compactness of the embedding V A c → L 2 (X) [To make a nervous reader happy, it suffices to show that if u n u in V A then C ε,λ (u n − u, v) → 0 as n → ∞; but this is immediate owing to the mean value theorem for f ∈ C 1 and the fact that f ε ∈ L ∞ (R)]. Thus, the generalized problem for Au ε + f ε (u ε ) = h can be written as A is bounded, continuous and pseudo-monotone, as the strongly continuous perturbation C ε,λ of the pseudo-monotone operator A λ (see [23,Proposition 27.6]). By (2.7), if 0 ≤ λ ≤ α 0 is sufficiently small then C ε,λ is coercive (indeed, C ε,λ (u, u) ≥const), and so is B ε since A λ is coercive by (2.10). Finally, the application of [23,Theorem 27.A] to the approximate problem yields the existence of at least one weak solution u ε ∈ V A . In particular, this means that for h ∈ L s (X) → V * A and u ε ∈ V A , the identity holds for all v ∈ V A . Here in (2.11), we have set f ε,λ (t) := f ε (t) − λt and assumed λ ∈ [0, α 0 ] with α 0 > 0 as in (2.7). Our subsequent goal is to show that u ε ∈ L ∞ (X) uniformly with respect to ε > 0. For u ε ∈ V A and k ≥ k 0 := t 0 , we define u ε,k := (|u ε | − k) + sgn(u ε ) and A k := {x ∈ X : |u ε (x)| > k}. Clearly, u ε,k ∈ V A with u ε,k = 0 on X\A k by Proposition 2.5. Taking v = u ε,k as a test function in (2.11) and exploiting (2.2), we find owing to the uniform bound (2.7) [Indeed, observe that f ε,λ (t) sgn (t) ≥ α, for |t| ≥ t 0 ]. From (2.12), Holder's inequality and the fact that V A → L 2q A (X), it follows for some constant C > 0 independent of ε; here, s, p > 1 are such that 1/2q A + s −1 + p −1 = 1. We further recall that by assumption (2.4) we have for every k ≥ k 0 , 14) The first inequality on the left-hand side of (2.14) is a consequence of the embedding V A → L 2q A (X). Let now h > k and observe that A h ⊂ A k , and on A h we have |u ε,k | ≥ h − k. This implies from (2.14) that Since 1/p = 1 − 1/ (2q A ) − 1/s > 1/ (2q A ) owing to the fact that s > q A / (q A − 1), we have p < 2q A , and therefore, ρ := 2q A /p > 1. Holder's inequality combined once again with (2.15) gives The application of Lemma 2.4 to (2.16) with Ξ(h) := χ A h L 2q A (X) yields the existence of a constant C > 0 that is independent ε > 0, such that This yields that |u ε (x)| ≤ K, a.e x ∈ X and thus the desired uniform bound in L ∞ (X) . Summing up, we have obtained for some constant C > 0 independent of ε, h and u ε . Finally, we can combine (2.17) with the coercitivity estimate for By taking a subsequence if necessary, from (2.17), (2.18) we may infer that that there exists a function u ∈ V A ∩ L ∞ (X) such that as ε → 0 + , and is standard owing to the properties of the Dirichlet form E A . Therefore, we have obtained the desired bounded solution u ∈ V A ∩ L ∞ (X) in the sense of Definition 2.6. The proof is finished.
Finally, we state and prove a very important theorem about the nonnegative self-adjoint operator A that is in one-to-one correspondence to the Dirichlet form E A (see (2.1)).
The semigroup can be extended to a contraction semigroup on L p (X, m) for every p ∈ [1, ∞], and each semigroup is strongly The operator A has a compact resolvent, and hence has a discrete spectrum.
Proof. The first two conclusions are more or less consequences of the abstract theories of [6] and [5]. In particular, since E A is a Dirichlet form the operator −A generates a submarkovian semigroup (e −tA ) t≥0 on L 2 (X) which is also analytic (see [ the operator A has a compact resolvent. By [5,Corollary 2.4.3], the estimate (2.23) also implies that the semigroup (e −tA ) t≥0 is ultra-contractive in the sense of (2.22). Since supp(m) = X and m (X) < ∞, the compactness of the semigroup on L 2 (X) together with the ultra-contractivity estimate (2.22) implies that the semigroup on L p (X) is compact for every p ∈ [1, ∞] (see, e.g., [5, Theorem 1.6.4]). We briefly explain how to get the last two conclusions of the theorem. We first observe that if u n ∈ D (A) is an eigenfunction associated with λ n , i.e., Au n = λ n u n . Let 0 < υ ∈ ρ (−A), the resolvent set of −A, such that υI + A is invertible. Then u n = (υI + A) −1 (λ n + υ) u n and exploiting the following representation it follows owing to (2.22) that u n L ∞ (X) ≤ C (λ n + υ) u n L 2 (X) < ∞, which is the desired conclusion. Since υI + A is invertible we have that the L 2 (X)-norm of (υI + A) θ defines an equivalent norm on D A θ . Besides, for every f ∈ L 2 (X), Using (2.22) for t ∈ (0, 1) and the contractivity of e −tA for t > 1, for u ∈ D A θ , we deduce The first integral is finite if and only if θ > d A /4; hence, the fourth conclusion of the theorem is proven as well.
Assume that (X, m) is a σ-finite measure space and V A c → L 2 (X). We conclude this subsection with a Poincare-Wirtinger like inequality in the space V A in the case when λ 1 = 0 is an eigenvalue of A (and so 1 ∈ D (A) is an eigenfunction). Then, it holds for some C A,X > 0 independent of u. Here, we have set The general strategy of proof is based on a contradiction argument and the compactness of the embedding V A c → L 2 (X), so we choose the omit the details. Alternatively, (2.24) follows also as a special case of [24, Lemma 4.3.1]. As a consequence, we observe that both define also equivalent norms for V A and its dual V * A , respectively.

2.2.
Examples of Dirichlet forms. In this section, we assume that X = Ω is a bounded domain in R n , n ≥ 1 with Lipschitz continuous boundary ∂Ω, where m is the usual Lebesgue measure on Ω. We shall apply the statement of Theorem 2.9 to a variety of operators and the associated Dirichlet forms.
The boundary condition in (2.25) is understood in the following (variational) sense: [5,6]) and all the results (i.e., Theorem 2.7 and Theorem 2.9) of the previous section are indeed applicable. More precisely, we have V A → L 2n n−2 (Ω) , provided that n > 2, and V A → L r (Ω), for any r ∈ (2, ∞), in the case n ≤ 2. It holds q A N = n/ (n − 2) or q A N = r/2 according to whether n > 2 or n ≤ 2, respectively. In particular, the solution of the elliptic problem (2.3) enjoys all the properties stated by Theorem 2.7 provided that h ∈ L s (Ω) with s > n 2 if n > 2 and with s ≥ 1 + δ r if n ≤ 2, for some (arbitrarily) small δ r > 0. Finally, A = A ∆ N enjoys all the properties stated in Theorem 2.9 and in particular, since q A = n/ (n − 2) the embedding D (A) → L ∞ (Ω) holds provided that n < 4. The latter is also known to be optimal with respect to the assumption on Ω (see, for instance, [5]).
Next, we give examples of Dirichlet forms that are associated with proper fractional Laplace operators with various boundary conditions, that were introduced recently in [12,14,22] (cf. also [13]). To this end, suppose that we are given a (symmetric) kernel J (x, y) = J (y, x), where J : Ω × Ω → R satisfies the following condition: for two constants 0 < c Ω ≤ C Ω depending only on Ω and J. Here, we have let s ∈ (1/2, 1) and then put , with the latter being dense and compactly contained in L 2 (Ω).
Proposition 2.11. E A is a Dirichlet form in the sense of Definition 2.1. It also satisfies the conclusion of Proposition 2.5.
Proof. Most of the properties in Definition 2.1 can be checked directly for the form E A . We show that for a Borel version of u ∈ L 2 (Ω), that satisfies u = 0 a.e., it DOUBLY NONLOCAL CAHN-HILLIARD EQUATIONS 143 holds E A (u, u) = 0. For this, take a compact subset Ω c ⊂ Ω and define the sets on account of (2.27), u = 0 a.e in Ω, and the symmetry of J. In particular, we have so that passing to the limit first as ε → 0 + and then as Ω c Ω, we get Owing to V A c → L 2 (Ω) , u n converges strongly to some function u ∈ V A in L 2 (Ω) , along a proper subsequence u n k (which has a limit u (x) a.e. in Ω). Then, by Fatou's lemma. Passing now to the limit as m → ∞ on both sides of the preceding inequality, we deduce Then the conclusion of Proposition 2.5 is also verified. The proof is finished.
Example 2.12. Let now A be the (nonnegative) self-adjoint operator that is found into a one-to-one correspondence via (2.1) to the Dirichlet space (E A , V A ), as given by (2.28). The statement of Theorem 2.9 is applicable to A. Observe that V A → L 2n n−2s (Ω) whenever n > 2s, while V A → L r (Ω) in the case n ≤ 2s, we have q A = n/ (n − 2s) for n > 2s, as well as, q A = r/2 whenever n ≤ 2s. In particular, the embedding D (A) → L ∞ (Ω) is also verified provided that n < 4s. The statements of Theorem 2.7 and Corollary 2.8 also apply to the elliptic boundary value problem (2.3) associated with the operator A.
Unfortunately, under the general assumptions of Proposition 2.11 there is no further explicit characterization of the operator A (of Example 2.12) and its domain D (A) in the literature. However, in some special situations when the kernel J is more explicit such characterizations can be proven (see [12,14,22]). In order to recall this case in detail, let us fix s ∈ (0, 1) , and set We have L(Ω) = ∅ since it contains L ∞ (Ω) and also smooth functions with compact support in Ω. Then, for u ∈ L(Ω), we define the regional fractional Laplacian (−∆) s Ω u by the formula provided that the limit exists, with the normalized constant C n,s as Next, suppose that Ω ⊂ R n is a bounded domain of class C 1,1 with boundary ∂Ω. We define the following boundary operator.
provided that the limit exists.
β (Ω) be the representation of u. Let u 0 := f h β so that u = u 0 + g. Then the following assertions hold.
1. If β ∈ (1, 2), then for z ∈ ∂Ω, (2.31) 2. If β = 2, then for z ∈ ∂Ω, Next, let and let the constant B n,s be such that We have the following fractional Green type formula for the regional fractional Laplace operator (−∆) s Ω (see [14]). Theorem 2.15. Let 1/2 < s < 1 and let (−∆) s Ω be the nonlocal operator defined in (2.29). Then, for every u : In this instance, for u ∈ C 2 2s (Ω) the function B n,s N 2−2s u is called the fractional normal derivative of u in direction of the outer normal vector.
Let us now return to the case when Ω ⊂ R n is a bounded domain with Lipschitz continuous boundary ∂Ω. We may consider the bilinear symmetric closed form E N with domain D(E N ) = H s (Ω) and given for u, v ∈ H s (Ω) by Since H s (Ω) = H s 0 (Ω) for all 0 < s ≤ 1/2 , we shall further assume that 1/2 < s < 1 in order to have a nontrivial boundary "trace" u |∂Ω = 0, and boundary condition for u (see, for instance, [22] for further details). Note once more that E N is a special case of (2.28). According to Proposition 2.11 (cf. also [22]), E N is a Dirichlet form on V A = H s (Ω) , while A = A N is the closed linear self-adjoint operator which can be defined in the sense of (2.1). We may call A N as the realization of the regional fractional Laplace operator (−∆) s Ω on L 2 (Ω) with the fractional Neumann type boundary conditions B n,s N 2−2s u = 0 on ∂Ω. Indeed, we have the following explicit description of the operator A N proven in [ All the considerations of Example 2.12 apply to the operator A N (recall that s ∈ (1/2, 1)).
We conclude this section with some important examples of Dirichlet forms defined over compact manifolds without boundary.

CIPRIAN G. GAL
We observe that the integration by parts formula holds: For the closed bilinear symmetric form we observe that 3. The abstract parabolic problem. Let us first introduce the initial value problem that we wish to investigate. In this section, assume (X, m) is σ-finite measure space. Consider subject to the initial condition u |t=0 = u 0 in L 2 (X) .
Finally, assume that λ 1 = 0 is an eigenvalue of A (i.e., 0 ∈ σ d (A)), and suppose that constant (in x ∈ X) functions belong to D (B) (i.e., 1 ∈ D (B)). We note that the latter condition 1 ∈ D (B) is actually not a restriction but in fact the most interesting case to investigate, while the former one (0 ∈ σ d (A)) is necessary in order to have the following property for (3.1)-(3.2): As far as the nonlinear potential f = F is concerned, we make the following assumptions: (Hf-2) There exists a constant c f > 0 and p ∈ (1, 2] such that |f (s)| p ≤ c f (|F (s)| + 1) , for all s ∈ R.
(Hf-3) There exist C 1 > 0, C 2 ≥ 0 and p ∈ (1, 2] such that Remark 3.1. The following remarks are in order. • (Hf-1) implies that F is a quadratic perturbation of some strictly convex function; (Hf-2) is satisfied by potentials of arbitrary polynomial growth of order p = p/ (p − 1). For instance, the double-well potential F (s) = θs 4 −θ c s 2 with 0 < θ < θ c , satisfies both (Hf-2) and (Hf-3) with p = 4/3. • Under assumption (HA-B), we note that the whole statement of Theorem 2.9 is in full force. In particular, A and B can be any of the operators as suggested by Examples 2.10, 2.12, 2.18 and 2.19.
3.1. Well-posedness: Weak and strong solutions. Let us define the (weak) energy space for some given M > 0, and equip Y M with the following metric Our definition of a weak solution for the abstract boundary value problem (3.1)-(3.3) is as follows.
Definition 3.2. Let u 0 ∈ Y M and 0 < T < +∞ be given. We say u is a weak solution if • We have u(0) = u 0 in X. We also define what we mean by a strong solution. To this end, we define a strong energy space where µ 0 is computed via the equation Here we have also equipped D (B) with its graph norm.
Definition 3.4. Let 0 < T < +∞ be given. We say u is a strong solution of (3.1)-(3.3) if it is a weak solution in the sense of Definition 3.2, and u, µ satisfy In particular, for the strong solution we have ∂ t u = −Aµ, a.e. in X × (0, T ) and µ = Bu + f (u), a.e. in X × (0, T ).
We first prove the existence of a strong energy solution to problem (3.1)-(3.3) by passing to the limit as , α → 0 in a regularized version of the system (see below). The latter possesses a sufficiently smooth solution u ,α . After that we will derive additional uniform estimates for the solutions u ,α as , α → 0 in order to pass to the limit. Remark 3.6. Note that by Theorem 2.9, q B > 2 is equivalent to having D (B) → L ∞ (X), and therefore for any u ∈ L ∞ (0, T ; Z M ) , the nonlinear term f (u) ∈ L ∞ (0, T ; L ∞ (X)) is well-defined. Proof. Step 1. As usual, one takes for ∈ (0, 1), a family of smooth functions {f } such that f → f uniformly on compact intervals of R, with the property that |f (s) | ≤ c f, , for all s ∈ R, and some constant c f, > 0 which may explode as → 0. Since F (s) = s 0 f (ζ)dζ, one may additionally assume that lim sup →0 F (s) = F (s) , (3.12) such that for all s ∈ R, owing to assumption (Hf-1) (cf. also Remark 3.1). We then insert an additional term α∂ t u, α > 0, into (3.2). More precisely, our approximated problem P ,α , , α ∈ (0, 1), consists in finding a function u ,α ∈ H 1 0, T ; L 2 (X) , in the regularity class of (3.10)-(3.11), to the abstract problem subject to the initial condition We also prepare a fixed initial datum µ ,α (0) ∈ L 2 (X) that solves Observe that the latter also yields that µ ,α (0) ∈ V A uniformly with respect to , α > 0 as µ 0 ∈ V A . In fact, from (3.17), we have so that E A µ ,α (0) , µ ,α (0) < ∞, uniformly in ( , α) provided that µ 0 ∈ V A . Observe in (3.17) that also µ ,α (0) X = µ 0 X and so then, clearly, for some C > 0 independent of α, . For u 0 ∈ D (Z M ), the existence of at least one strong solution to problem P ,α , in the sense of Definition 3.4, follows from the application of a backward Euler scheme [7, Theorem 2.9 and Remark 2.1].
Step 2. Now, we derive uniform estimates for u ,α with respect to , α ∈ (0, 1). We point out that all test functions used in this step are admissible in working with (3.14)-(3.15) on account of the regularity of u ,α and µ ,α . For practical purposes, in this proof C, Q > 0 denote a positive constant and a function that are independent of α, and u. Such parameters may vary even from line to line. Further dependencies of the constants and functions on other parameters will be pointed out as needed.
for every ε > 0. Inserting the foregoing inequality into (3.29), for a sufficiently small ε ∈ (0, 2C), we get The application of Gronwall inequality yields once again (3.31)-(3.32). By virtue of (3.14)-(3.15), we note that (3.31) also automatically gives as well as where in the right-hand side of inequality (3.31) we have actually inserted (3.18). In order to derive an uniform bound for µ ,α in L ∞ (0, T ; V A ) we test (3.15) by u ,α := u ,α − u 0 X and recall that u ,α X = u 0 X (clearly, then u ,α X = 0). We find Since µ ,α , u ,α = Aµ ,α , A −1 u ,α = A 1/2 µ ,α , A −1/2 u ,α , the first term on the right-hand side of (3.36) can be estimated in terms of for any ε > 0. Clearly, as 1 ∈ D (B) by assumption, we have for some C M = C M (f, g, ϕ 0 ) > 0 independent of ( , α). Thus, (3.37) allows one to deduce owing to the fact that ∂ t u ,α (t) X = 0 and the self-adjointness of B. Thus, since all the terms on the right-hand side of (3.39) are already estimated in (3.38), and since 1 ∈ D (B) , we further infer by virtue of (3.21) that Consequently, from equation (3.14) we deduce Collecting now (3.40), (3.34) together with (3.35), we deduce that In order to deduce the uniform L ∞ (X)-estimate for u ,α , we exploit the "elliptic" regularity result of Theorem 2.7 (owing to the fact that s = 2 > q B / (q B − 1) ⇔ q B > 2) for the nonlinear problem: Bu ,α + f (u ,α ) = h ,α := µ ,α − α∂ t u ,α , m-a.e in X, (3.45) where α ∈ (0, 1). Indeed, the preceding estimates (3.42)-(3.44) allow us to deduce (3.46) Therefore, by (3.13) and (3.46) we can infer Finally, having obtained (3.47) we can interpret the nonlinearity f (u ,α ) in (3.45) as a bounded external force (note that we may assume that |f (s)| ≤ |f (s)|, for sufficiently large |s| ≥ s 0 , as both f , f are linear perturbations of some strictly monotone functions). We deduce according to the application of Corollary 2.8, Step 3. In this final step, we can finally pass to the limit as ( , α) → (0, 0) in the sequence of solutions u ,α satisfying the approximated problem P ,α . Recalling all the foregoing estimates, we deduce, up to subsequences, that On the other hand, due to (3.50)-(3.51) and classical compactness theorem of Aubin-Lions-Sobolev, we also have On account of (3.55), (3.51), We now have all the ingredients to deduce that u is a strong solution to (3.1)-(3.3) on (0, T ), for any T > 0. Indeed, we can pass to the limit in the approximating problem for u ,α , and find that u is a strong solution such that u ∈ L ∞ (0, T ; Z M ). (3.50), it also follows that µ ∈ L 2 (0, T ; D (A)). Finally, we recover the continuity property u(0) = u 0 in X by means of a standard argument (note for instance (3.55)). The proof is finished.
We have the following existence of weak solutions for problem (3.1)-(3.3). Proof. Choose a smooth sequence of data u 0n ∈ Z M such that u 0n → u 0 ∈ Y M , i.e., u 0n → u 0 in V B and F (u 0n ) → F (u 0 ) in L 1 (X) (here we can use implicitly that D(B) is dense in V B ). The corresponding strong solution u n exists by Theorem 3.5. Arguing as in the the proof of (3.19)-(3.20), we find d dt E n (t) + E A (µ n (t) , µ n (t)) = 0, (3.57) for all t ≥ 0, where we have set As before, owing to the fact that F is a quadratic perturbation of a strictly convex functions (see (Hf-1)), we first observe Next, let us recall the identity (3.39) which actually holds by any smooth solution u n , . We have the following cases: . The foregoing estimates (3.59)-(3.66) and on account of the compact embedding we infer the existence of such that, as n → ∞, Thus, on account of (3.69) we can pass to the limit as n goes to infinity in the problem satisfied by the strong solution u n . We easily obtain that the limit function in (3.67) is the desired weak solution in the sense of Definition 3.2.

and so by duality and compactness
Also in this case, the compactness of the embedding yields the desired properties in (3.68), as well as (3.67), (3.69). Therefore, the argument leading to the desired final conclusion is the same. This concludes the proof.
We now verify the stability of solutions of problem (3.1)-(3.3) with respect to the initial data in Y M . Theorem 3.8. Let u i be any two weak solutions in the sense of Definition 3.2, corresponding to the initial data u i (0) , i = 1, 2, with u 1 (0) X = u 2 (0) X . Assume (Hf-1) and that u i ∈ L ∞ 0, T ; L p (X) for any T > 0. Then the following estimate holds: where C > 0 is independent of u i , time and the initial data.
Proof. Set (µ, u) such that µ := µ 1 − µ 2 and u := u 1 − u 2 . Then (µ, u) satisfies the system subject to the initial condition By assumption, we also have u (0) X = 0 and so u (t) X ≡ 0, as well. Testing equation (3.71) by A −1 (u) and equation (3.72) by u, we deduce Recalling the second condition of (Hf-1), we get for some constant C = C B > 0 depending only on X and B. We can now estimate the term on the right-hand side of (3.75) exactly as in the cases (a)-(b) of the proof of Theorem 3.5 (see (3.30)-(3.35)) and then apply the Gronwall inequality. The proof is finished.
Remark 3.9. The assumption q B > 2 on the operator B in the statement of Theorem 3.7 is inessential. Indeed, in the proof of Theorem 3.7 we can deal instead with a class of strong solutions u ε that satisfy the following parabolic problem: for some operator D which satisfies (HA-B) with q D > 2, such that D (D) ⊆ D (B) and 0 ∈ σ d (D) (assuming also for simplicity that 0 ∈ σ d (B)). The existence of a strong solution u ε to (3.76) can be handled exactly as in the proof of Theorem 3.5. Then one may pass to the limit as ε → 0 in (3.76) to obtain the desired weak solution without that assumption. This procedure may be tailored according to the specific situation one is interested in.
We omit the simple (repetitive) details, but leave them to the interested reader and conclude this subsection with a result that combines all the previous statements.

Finite dimensional attractors.
In this subsection, we shall establish the existence of an exponential attractor for problem (3.1)-(3.3). Our first result is concerned with a dissipative estimate enjoyed by the (unique) weak solution of Theorem 3.7. We require an additional (compare to the second of (Hf-1)) weak assumption on F as follows.
(Hf-4) There exist sufficiently large constants C 1 > 0, Here, C A,X > 0 is the best Poincare-Wirtinger constant in the inequality , u ∈ V A , and c F > 0 is the constant from (Hf-1).

Let us set
In the rest of the (sub)section, we shall proceed formally when performing asymptotic estimates, but recall that the arguments can always be made rigorous by choosing to work with strong solutions (see Theorem 3.5 and Remark 3.9). The clean estimates follow by a standard passage to the limit in these uniform estimates for the strong solutions. Proof. We first note that (3.78) is an immediate consequence of (3.77). To show (3.77) we can proceed formally. In particular, each weak (and strong) solution satisfies the following energy identity for all t ≥ 0. Next, let us test (3.2) with u. We obtain By the assumption (Hf-1) owing to the fact that F is a quadratic perturbation of a strictly convex function, for all s ∈ R we have Therefore, from (3.80) we get On the other hand, we can exploit the Poincare-like inequality µ − µ X 2 L 2 (X) ≤ C A,X E A (µ, µ), and the conservation of mass u X = u 0 X , to observe that assuming for simplicity (for now) that u 0 X (= u X )= 0. Thus, by virtue of assumption (Hf-4) we rewrite (3.81) and continue to estimate as follows: Application of Young's inequality on the left-hand side of (3.82) yields and so we easily deduce for some constant L * > 0 which depends only on C 1 , F (0) and m (X) = |X| < ∞. It follows by virtue of (3.83) and the energy identity (3.79) that we have for all t ≥ 0. By means of the Gronwall inequality we then obtain Here t * > 0 is the same as in the statement of Theorem 3.11.
Proof. As usual, in this proof C, Q > 0 will denote a constant and function which are independent of u, time and the initial data. Let t * > 0 be the entering time from (3.78). For t ≥ t * , from (3.79) it follows that .
Integrating the foregoing identity over (t * , T ), with any T > t * , in light of (3.78) we obtain Moreover, the energy identity (3.79) also gives in light of (3.78) that Next, we recall (3.28)-(3.29) contained in the proof of Theorem 3.5. Both of these actually hold by any strong solution (of course, now α = 0). In particular, for all , for some constant C B > 0 which depends only on B and m (X) < ∞.
We once again need to deal with the cases when either (a) • (a) Inequality (3.89) yields in light of the same argument of (3.30) that (3.90) Multiplying both sides of (3.90) by (t − t * ) and integrating the resulting inequality over (T, t * ) , with any T ≥ t * + 1, we deduce on account of estimates (3.87)-(3.88) and the fact that ∂ t u (3.92) It remains to note that integrating (3.90) over (t, t + 1), and using (3.92) we also obtain (3.93) • (b) The case V B → V A may be argued exactly as in (3.33). This implies once again that (3.90) holds, and so both (3.92)-(3.93) can be verified as well.
Next, we exploit the fact that u ∈ L ∞ (t * , ∞; V B ) uniformly with respect to time and the initial data (see (3.78)) to observe that f (u) = F (u) ∈ L ∞ t * , ∞; L 1 (X) , uniformly in time and the initial data by virtue of the assumptions on F (see (Hf-2), (Hf-3)). Recalling from (3.39) that by application of (3.92) we find that µ X ∈ L ∞ (t * + 1, ∞) uniformly in time and the initial data. Together with (3.92) and the Poincare-Wirtinger inequality (2.24) this yields sup for some function Q > 0 independent of u, time and the initial data. In order to deduce the uniform L ∞ (X)-estimate for u (t), we can apply Theorem 2.7 to problem Bu (t) + f (u (t)) = µ (t), a.e. in X × (t * + 1, ∞), to obtain by virtue of (3.94) and (Hf-1) that sup Note that in this instance, h := µ ∈ V A → L 2q A (X) and so h ∈ L s (X) with s = 2q A > q A / (q A − 1) for as long as q A > 1 (the latter holds by assumption (HA-B)). It remains to exploit (3.95) and the statement of Corollary 2.8 with a nonlinearity which is to be interpreted as a bounded external force. We immediately arrive at sup  x − y V s,l is the Hausdorff semi-distance.
In order to prove this result we report the following basic abstract result on the existence of exponential attractors for discrete maps (see, for instance, [18]). Theorem 3.14. Let X 1 and X 2 be two Banach spaces such that X 2 is compactly embedded in X 1 . Let X 0 be a bounded subset of X 2 and consider a nonlinear map Σ : X 0 → X 0 satisfying the smoothing property

98)
for all x 1 , x 2 ∈ X 0 , where d > 0 depends on X 0 . Then the discrete dynamical system (X 0 , Σ n ) possesses a discrete exponential attractor E * M ⊂ X 2 , that is, a compact set in X 1 with finite fractal dimension such that where d X and ρ * are positive constants independent of n, with the former depending on X 0 .
Let us now introduce a ball B R ⊂ D(B) with sufficiently large radius R > 0 such that By Theorem 3.12, there exists a sufficiently large R M > 0 such that the ball B := B R M will be absorbing for S (t) and S (t) B ⊂ B for all t ≥ t * + 2. Thus it suffices to construct the exponential attractor on the set B. Next, we prove a series of elementary lemmas for which Theorem 3.12 proves essential. The first lemma is concerned with the Lipschitz-in-time regularity of the semigroup S(t).

102)
for all t ≥ 1, and some positive constant C which only depends on B.
Proof. First, we note that due to estimate (3.86) we have Here the constant R 0 > 0 is independent of time and depends only on B. In this proof, the constant C = C (R 0 ) > 0 will be independent of time and the initial data. As in the proof of Theorem 3.8, set µ := µ 1 − µ 2 and u := u 1 − u 2 . Then (u, µ) satisfies the problem ∂ t u + Aµ = 0, (3.105) which is the analogue of (3.74) in the case when u (0) X = 0. Since by (3.103) we have a uniform estimate on the L ∞ -norms of the solutions u i , we can estimate the term on the right-hand side of (3.107), Since also for a strong solution u ∈ D (B), for every δ > 0. Arguing in (3.109) as before, according to whether (a) V A → V B or (b) V B → V A (see the proof of Theorem 3.5), by Gronwall inequality we get Next, we test (3.105) by A −1 (∂ t u) and (3.106) by ∂ t u, and obtain after standard transformations (note that this time For the first term on the right-hand side, we can use (3.103) to find Inserting now these estimates on the right-hand side of (3.111), we get We now need to control the last term on the right-hand side of (3.113). To this end, we differentiate both equations of (3.105)-(3.106) with respect to time, and set v := ∂ t u (= ∂ t u) and w := ∂ t µ. Then (v, w) solves the problem We note that v (t) X = 0. Next, we test (3.114) with A −1 (v) and (3.115) with v.
Adding the resulting equations, we obtain owing to assumption (Hf-1) and (3.103) to control the L ∞ -norm. Here, we have also estimated directly when V B → V A , whereas for V A → V B we have exploited Proposition 2.3. We have also used the fact that |u| |∂ t u 2 | ∈ V * B as a product of functions in L 2 (X) × V B (see assumption (3.97)). Indeed, for sufficiently smooth by Holder's inequality. Inserting now (3.117) into the right-hand side of (3.116) we can infer the existence of a sufficiently small δ 0 > 0 such that The idea now is to add (3.113) to (3.118) and to absorb the term ∂ t u (t) 2 L 2 (X) from the right-hand side into the δ 0 -term on the left-hand side of (3.118) for some sufficiently small δ < δ 0 (dealing with each of the cases for all t > 0. Multiplying now both sides of (3.119) by t > 0, we have We note preliminarily that by virtue of (3.104) and (3.110) we have (3.121) Integrating now (3.120) over (0, T ) with any T ≥ 1, we derive This entails, owing to the first of (3.121), that On the other hand, the application of Gronwall's inequality to (3.122) yields owing once again to (3.121). Since C u (T ) 2 V B ≤ F (T ) the claim (3.102) follows from (3.123) and this completes the proof.
Proof of Theorem 3.13. We shall now exploit the statements of Lemmas 3.15 and 3.16 to finish the proof. First, we recall once again that B := B R M is an absorbing set for S (t) and S (t) B ⊂ B for t ≥ t , for some t ≥ 1. Thus we can apply the statement of Theorem 3.14 to the map Σ = S(t ). Indeed, setting X 0 = B the mapping Σ : X 0 → X 0 enjoys the smoothing property (3.102) owing to the fact that the embedding V B c → L 2 (X) is compact. Therefore Theorem 3.14 applies to Σ and there exists a compact set M * ∈ X 0 with finite fractal dimension (with respect to the metric topology of L 2 (X)) that satisfies (3.99) and (3.100). Moreover, since B is an absorbing set for these semigroups in Y M then these (discrete) attractors attract exponentially fast all the bounded subsets of Y M . In order to construct the attractor for the semigroup with continuous time, as usual we use the formula We now recall that the semigroup S (t) is Lipschitz continuous on [t , 2t ] × B in the norm of V * A . This was verified in Lemma 3.15 as well as in Theorem 3.8 (cf. also (3.110)). Thus, we can verify in a standard way that the exponential attractor M satisfies (i), (ii) and (iii) of Theorem 3.13, but with the norm of V B being replaced by that of V * A . In order to extend these properties to the required V B -norm it suffices to recall that M is bounded in B, the semigroup possesses a smoothing property from Y M into Z M and to argue as before according to the cases whether V A → V B or V B → V A , as well as the fact that D(B) → V B → L 2 (X). The proof of Theorem 3.13 is finished.
As a consequence of Theorem 3.13, we may conclude this section with the following. Remark 3.18. We note that assumption (3.97) of Theorem 3.13 is always satisfied if q B > 2, owing to (HA-B) and the fact that L 2q B (X) → L 2q B /(q B −1) (X) in that case.
3.3. Remarks on the singular potential case. We conclude the paper with a well-posedness result for the doubly nonlocal problem (1.5) with a potential F that may be logarithmic-like. More precisely, we suppose that is a globally Lipschitz-bounded function such that f 1 ≥ −θ c . In applications, very often f 1 (s) = −θ c s and f 0 (s) = θ ln 1+s 1−s , for some θ c > θ > 0. We can prove the following.
It may be possible to prove additional regularity results for our doubly nonlocal problem in the case when F is a singular logarithmic potential, but we feel that this case would deserve a separate and complete investigation. Our goal in Theorem 3.19 was merely to show that our framework can be easily extended to deal with this important case as well. In particular, our result in Theorem 3.19 stands for a clear generalization of some well-posedness results derived in [1] where specifically, A = −∆ Ω,N , B is the self-adjoint operator given by Example 2.12 (cf. Subsection 2.2).