An asymptotic analysis for a generalized Cahn-Hilliard system with fractional operators

In the recent paper `Well-posedness and regularity for a generalized fractional Cahn-Hilliard system' (Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30 (2019), 437-478 -- see also arXiv:1804.11290), the same authors have studied viscous and nonviscous Cahn-Hilliard systems of two operator equations in which nonlinearities of double-well type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers $A^{2r}$ and $B^{2\sigma}$ (in the spectral sense) of general linear operators $A$ and $B$, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space $L^2(\Omega)$, for some bounded and smooth domain $\Omega\subset{\mathbb{R}}^3$, and have compact resolvents. Existence, uniqueness, and regularity results have been proved in the quoted paper. Here, in the case of the viscous system, we analyze the asymptotic behavior of the solution as the parameter $\sigma$ appearing in the operator $B^{2\sigma}$ decreasingly tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of $B$ appears.


Introduction
A research project that the three of us recently carried out in [15][16][17], deals with the well-posedness, regularity and optimal control for the abstract evolutionary system where A 2r and B 2σ , with r > 0 and σ > 0, denote fractional powers of the linear operators A and B, respectively. These operators are supposed to be densely defined in H := L 2 (Ω), with Ω ⊂ R 3 , selfadjoint and monotone, and to have compact resolvents. The above system is a generalization of the standard or viscous Cahn-Hilliard system (depending on whether τ = 0 or τ > 0), which models a phase separation process taking place in the container Ω. The particular sample case A 2r = B 2σ = −∆ with homogeneous Neumann boundary conditions is included, indeed. The physical variables ϕ and µ stand for the order parameter and the chemical potential, respectively, while f is a given source term. Moreover, F denotes a double-well potential. We offer three physically significant examples for F , namely, where the constants c i in (1.5) and (1.6) satisfy c 1 > 1 and c 2 > 0, so that F log and F 2obs are nonconvex. These potentials are called the classical regular potential, the logarithmic potential , and the double obstacle potential, respectively. In irregular situations like (1.6), one has to split F into a nondifferentiable convex part β (the indicator function of [−1, 1], in the case of (1.6)) and a smooth perturbation π. At the same time, one has to replace the derivative of the convex part by the subdifferential and to interpret (1.2) as a differential inclusion or, equivalently, as a variational inequality involving β rather than its subdifferential, as actually done in [15].
Fractional versions of the Cahn-Hilliard system have been considered by different authors and are the subject of several papers. As for references regarding well-posedness and related problems, a rather large list of citations is given in [15]; we recall some concerned and recent literature also here, by mentioning [1,2,8,21,30,33]. Moreover, one can find a number of results regarding the asymptotic behavior of solutions, for the standard Cahn-Hilliard equations, for variants thereof, and for systems including the Cahn-Hilliard equations: without any claim of completeness, we can quote, e.g., [3, 6, 9-12, 14, 18-20, 23-25, 31, 32, 34, 35]. These works mainly deal with the asymptotics with respect to parameters, or the study of the trajectories and related topics, or the existence of global or exponential attractors and their properties. A special role in our citations is played by the paper [13], where the longtime behavior of the solutions as well as an asymptotic analysis similar to the one we address here are investigated for a fractional system involving the Allen-Cahn equation.
In this paper, we consider the viscous case τ > 0 within the system (1.1)-(1.3) and study the asymptotic behavior of the solution as the parameter σ involved in the operator B 2σ tends to zero. In this analysis, a crucial role is played by the orthogonal projection operator P : H → H on the kernel ker B of B. Indeed, if (ϕ σ , µ σ ) denotes the solution to system (1.1)-(1.3) for an arbitrary σ > 0, we prove that (ϕ σ , µ σ ) converges as σ ց 0 to a solution (ϕ, µ) to the system (1.9) In general, the convergence occurs along a subsequence, but in the case when the limit pair (ϕ, µ) uniquely solves (1.7)-(1.9), then the whole family (ϕ σ , µ σ ) converges to (ϕ, µ) in the sense made precise by the statement of Theorem 2.5 below. Moreover, let us point out that the component ϕ of the pair (ϕ, µ) is always uniquely determined, as it follows from the continuous dependence result given by Theorem 2.10. In the last part of the paper, we also discuss the limiting problem by proving a class of regularity results, quite interesting in our opinion, for which we have to use some sophisticated tools of interpolation theory. Our approach may be considered as an extension and further investigation with respect to the asymptotic results of [13,Section 7], in which a phase relaxation problem is obtained at the limit. Also in the present paper the equation (1.8) can be seen as an ordinary differential equation, but with a nonlocal structure due to the presence of the projection operator P . Our contribution here gives account of a new line of investigation that in our opinion should be further explored.
The rest of the paper is organized as follows: in the next Section 2, we list our assumptions and state our results. The corresponding proofs are given in the last two Sections 3 and 4.

Statement of the problem and results
In this section, we state precise assumptions and notations and present our results. Our framework is the same as in [15], and we briefly recall it here, for the reader's convenience. First of all, the open set Ω ⊂ R 3 is assumed to be bounded, connected and smooth. We use the notation H := L 2 (Ω) (2.1) and denote by · and ( · , · ) the standard norm and inner product of H. As for the operators involved in our system, we postulate that and (e i , e j ) = (e ′ i , e ′ j ) = δ ij for i, j = 1, 2, . . . , where δ ij denotes the Kronecker index. The power A r of A with an arbitrary positive real exponent r is given by In principle, we could endow V r A with the standard graph norm in order to make V r A a Hilbert space. However, we will choose an equivalent Hilbert structure later on. In the same way, for σ > 0, we define the power B σ of B. For its domain we use the notation At this point, we can start listing our assumptions. First of all, r, σ 0 and τ are fixed positive numbers, and σ ∈ (0, σ 0 ) is a parameter. (2.8) As for the linear operators, we postulate, besides (2.2), that either λ 1 > 0 or 0 = λ 1 < λ 2 and e 1 is a constant; (2.9) if λ 1 = 0, then the constant functions belong to V σ B . (2.10) In [15] some remarks are given on the above assumptions. Moreover, it is shown that an equivalent Hilbert structure on V r A is obtained by taking the norm defined by 11) and the corresponding inner product, which we term ( · , · ) A,r . This equivalence is trivial if λ 1 > 0. In the opposite case λ 1 = 0, with the notation for the mean value ot the generic function v, the equivalence relies on the inequality which is of Poincaré type, since the term (v, e 1 ) appearing in (2.11) and involving the constant function e 1 (see (2.9)) is proportional to mean v. Next, the nonlinear potential F appearing in (1.2) is split as follows: where (2.14) β : R → [0, +∞] is convex, proper and l.s.c. with β(0) = 0; (2.15) π : R → R is of class C 1 with a Lipschitz continuous first derivative; (2.16) it holds lim inf |s|ր+∞ s −2 F (s) > 0 . (2.17) Notice that these assumptions are fulfilled by all of the important potentials (1.4)-(1.6). We set, for convenience, β := ∂ β, π := π ′ , and L π := the Lipschitz constant of π .
(2.18) Moreover, we term D( β) and D(β) the effective domains of β and β, respectively, and notice that β is a maximal monotone graph in R × R. The same symbol β is used for the maximal monotone operators induced in L 2 (Ω) and L 2 (Q). Finally, we introduce P : H → H, the orthogonal projection operator on the kernel of B.
As for the data of our problem, we allow the forcing term appearing in (1.2) to depend on σ and assume that: and β(ϕ 0 ) ∈ L 1 (Ω); (2.21) if λ 1 = 0 then m 0 := mean ϕ 0 belongs to the interior of D(β). (2.22) At this point, we make the notion of solution precise. In the following, we use the notations Q t := Ω × (0, T ) for t ∈ (0, T ] and Q := Q T . (2.23) A solution to our system is a pair (ϕ σ , µ σ ) fulfilling the regularity requirements and satisfying the following weak formulation of the equations (1.1)-(1.3): for every v ∈ V σ B and for a.a. t ∈ (0, T ), (2.28) ϕ σ (0) = ϕ 0 . (2.29) We notice that (2.26) implies that β(ϕ σ (t)) ∈ L 1 (Ω) for a.a. t ∈ (0, T ), so that (2.28) has a precise meaning. In the same inequality, one obviously has to read Ω β Remark 2.1. The regularity (2.25) of the second component of the solution is expected even though (2.27) just suggests µ σ ∈ L 2 (0, T ; V r A ). Indeed, for a.a. t ∈ (0, T ) the variational equation has the form From this, one easily derives that u ∈ V 2r A and A 2r u ≤ g (one can formally test by A 2r u, but a regularization procedure makes the argument rigorous). Since ∂ t ϕ σ ∈ L 2 (0, T ; H), we thus have the regularity (2.25) as well as i.e., the equation holds in its strong form.
Remark 2.2. In the sequel, the symbol 1 denotes the constant function on Ω that takes the value 1 at every point. With this notation, we remark that (2.9) implies that A r (1) vanishes if λ 1 = 0, so that (2.27) and (2.29) imply that Remark 2.4. No uniqueness for the component µ σ of the solution can be expected, in general. However, in particular situations, µ σ is unique, too. This is the case if λ 1 > 0. Indeed, this assumption implies that A 2r is invertible so that (2.30) can be uniquely solved for µ σ . On the contrary, the case λ 1 = 0 is much more delicate. A sufficient condition that ensures uniqueness for µ σ is the following (see [15,Rem. 4.1]): β is an everywhere defined C 1 function and ϕ σ is bounded. We notice that the same argument used in the quoted remark also applies if D(β) is an open interval and β is a continuous single-valued function on it (like in the case (1.5) of the logarithmic potential) provided that all of the values of ϕ σ belong to a compact subset of D(β).

(2.36)
Moreover, under the additional assumption, the following holds true: whenever (ϕ σ , µ σ ) is a solution to problem (2.27)-(2.29) for σ > 0 and (2.33)-(2.35) hold for some triplet (ϕ, µ, ζ), then ζ = ϕ − P ϕ and the pair (ϕ, µ) is a solution to the system for every v ∈ H and for a.a. t ∈ (0, T ), (2.40) Remark 2.6. The above statement looks a little involved. Besides the assumption (2.37) we are going to discuss in a while, we point out that that no uniqueness for the solution (ϕ σ , µ σ ) is required. On the contrary, if additional assumptions were made that guarantee uniqueness for (ϕ σ , µ σ ) (see Remark 2.4) and (2.37) were assumed, then the statement would look much simpler, namely: as σ tends to zero, the solution (ϕ σ , µ σ ) converges (in the sense of (2.33)-(2.34), possibly along a subsequence) to a solution (ϕ, µ) to problem (2.38)-(2.40). If, in addition, uniqueness holds for the solution (ϕ, µ) to the limiting problem, then the whole family {(ϕ σ , µ σ )} converges to (ϕ, µ) as σ tends to zero. Remark 2.7. As observed in the forthcoming Remark 3.3, if (2.37) is not assumed, a weaker conclusion can anyway be obtained: the variational inequality (2.39) is fulfilled by all the test functions v ∈ V σ 0 B . Indeed, it is stressed in the remark that assumption (2.37) is used in the proof of Theorem 2.5 just to extend to any v ∈ H the validity of (2.39) proved for test functions v ∈ V σ 0 B . Remark 2.8. So, if (2.37) is assumed, then every limiting pair (ϕ, µ) satisfies (2.39) with arbitrary test functions v ∈ H. This has the following important consequence: there exists some ξ satisfying ξ ∈ L 2 (0, T ; H) and ξ ∈ β(ϕ) a.e. in Q , then ξ belongs to L 2 (0, T ; H), equation (2.42) is satisfied, and (2.39) becomes But this exactly means that ξ(t) ∈ ∂ β(ϕ(t)) = β(ϕ(t)) for a.a. t ∈ (0, T ), i.e., the second condition in (2.41). If instead (2.37) is not assumed, then (2.39) is satisfied only for test functions v ∈ V σ 0 B , as said in Remark 2.7. Nevertheless, the definition (2.43) still yields ξ ∈ L 2 (0, T ; H) and implies that (2.42) is satisfied. However, in this case, (2.44) is only true for v ∈ V σ 0 B , and this means that for a.a. t ∈ (0, T ) the function ξ(t) belongs to the subdifferential of the function Notice that this subdifferential is a subset of the dual space (V σ 0 B ) * and might contain elements that do not belong to H (in the sense of the Hilbert triplet . Moreover, if a function u ∈ H belongs to such a subdifferential, then it is not clear whether it also belongs to the subdifferential in H (i.e., that of the function H ∋ v → Ω β(v)), so that we cannot conclude that ξ ∈ β(ϕ) a.e. in Q. About this matter, let us quote the paper [5] for related issues.
Remark 2.9. A sufficient condition for (2.37) to hold true is the following (satisfied in all of the concrete cases, at least if σ 0 is small enough): In order to construct the sequence {v n } for a given v ∈ H, we solve the Neumann boundary value problem Since v ∈ H, we have that v n ∈ H 2 (Ω) and thus v n ∈ V σ 0 B , by (2.45). Now, if we take z = v n in (2.46) and use the Cauchy-Schwarz inequality in the right-hand side, then we easily find that v n ≤ v and 1 n ∇v n 2 ≤ 1 n v 2 for all n ∈ N. (2.47) Hence, there are a subsequence {v n k } and some w ∈ H such that v n k → w weakly in H. Moreover, since 1 n ∇v n → (0, 0, 0) strongly in H × H × H by (2.47), we easily infer from (2.46) that w = v. A fortiori, by the uniqueness of the limit point, the entire sequence {v n } converges weakly in H to v. But then, by the weak sequential lower semicontinuity where the latter inequality follows from (2.47). We thus have v = lim n→∞ v n , and the uniform convexity of H yields that v n → v strongly in H. Now, denoting by β ε and β ε the Moreau-Yosida ε-approximations of β and β, respectively (see, e.g., [7, p. 28]), we account for the definition of the subdifferential β ε = ∂ β ε and the identity obtained by testing (2.46) by β ε (v n ) ∈ H 1 (Ω). We have that and we deduce that by letting ε tend to zero. This implies the inequality "≤" in (2.37). Since the opposite inequality clearly follows from the lower semicontinuity of the function z → Ω β(z) in H, we finally conclude for the validity of (2.37).
Notice that Theorem 2.3 ensures the existence of at least one solution to the limiting problem (2.38)-(2.40) with the regularity specified in (2.36). Our next result deals with partial uniqueness and continuous dependence of the solution. This will be proved in the last Section 4, which is devoted to the study of the limiting problem.
Theorem 2.10. Let the general assumptions on the structure be fulfilled, and assume that ϕ 0 satisfies (2.21). Morever, let f i ∈ L 2 (0, T ; H), i = 1, 2, be two choices of the forcing term f appearing in (2.39), and let with a constant C cd that depends only on τ , the Lipschitz constant L π , and T . In partic- In our final result, we require some regularity of the data and further assumptions on the structure that are satisfied in all of the concrete cases of interest, and we prove a regularity result. As a byproduct, we obtain a sufficient condition for the uniqueness of the second component µ of the solution. Sufficient conditions for uniqueness in a different direction are given in the forthcoming Remark 4.5.
Theorem 2.11. Let the general assumptions on the structure be fulfilled. In addition, assume that be a solution to problem (2.38)-(2.40). Then ϕ enjoys the further regularity

52)
and there exists some ξ satisfying (2.41)-(2.42). In particular, even the second component µ of the solution is unique if β is single-valued.
Throughout the paper, we widely use the Cauchy-Schwarz and Young inequalities, the latter in the form Moreover, in performing a priori estimates, we use the same small letter c for (possibly) different constants that depend only on the structure of our system but σ, and on the assumptions on the data. In particular, the values of c do not depend on the regularization parameter λ we introduce in the next section. On the contrary, some precise constants are denoted by different symbols (see, e.g., (2.13), where a capital letter with an index is used).

Asymptotic analysis
This section is devoted to the proof of Theorem 2.5. The construction of the solutions (ϕ σ , µ σ ) mentioned in the statement relies on a priori estimates on the solutions to a regularized problem, as done in [15] to solve problem (2.27)-(2.29) with a fixed σ. Hence, we briefly recall that regularization procedure. For λ > 0 (small enough if needed), let β λ be the Yosida approximation of β at the level λ (see, e.g., [7, p. 28]). The corresponding Moreau regularization β λ of β is thus given by since β λ (0) = 0 due to (2.15). Then, the regularized problem consists in looking for a pair (ϕ λ σ , µ λ σ ) satisfying the regularity requirements and solving the following system: We notice that the variational inequality (2.28) is replaced by the equality (3.3) in the approximating problem (since β λ is an everywhere defined Lipschitz continuous function). The existence part of Theorem 2.3 is proved by solving the above regularized problem (cf. [15,Thm. 5.1]) and showing that its solution (ϕ λ σ , µ λ σ ) converges as λ ց 0 (in a suitable topology, possibly just along a subsequence) to a pair (ϕ σ , µ σ ) which turns out to solve problem (2.27)- (2.29). This solution, where now σ is a varying parameter that we intend to approach zero, will be the good candidate for Theorem 2.5.
Before starting to estimate, it is worth observing that Remark 2.1 applies to both equations (3.2) and (3.3). This is obvious for the former. As far as the latter is concerned, one has to replace A and r by B and σ, respectively, and notice that β λ is Lipschitz . This justifies the last regularity condition for ϕ λ σ in (3.1) (in contrast with (2.24)) and implies the strong form of both equations, i.e., We also recall the convention on the symbol c for possibly different constants made at the end of Section 2. Moreover, since (2.32) implies that f σ is bounded in L 2 (0, T ; H), we allow c to also depend on a bound for the corresponding norm.
First a priori estimate. We test (3.2) written at the time s by µ σ (s). At the same time, we insert +ϕ λ σ (s) to both sides of (3.6) written at the time s and multiply it by ∂ t ϕ λ σ (s), then integrating over Ω. We sum up both equalities, noting that a cancellation occurs, and integrate over (0, t) with respect to s. We obtain Even the last integral on the left-hand side is nonnegative. We estimate the terms on the right-hand side by accounting for the assumptions (2.21) and (2.32) on ϕ 0 and f σ , respectively, and owing to the Lipschitz continuity of π. Recalling also (2.7), we have that Therefore, by rearranging and applying the Gronwall lemma, we conclude that Second a priori estimate. Our aim is to improve the estimate concerning µ λ σ . Indeed, for the following we need that µ λ We notice at once that this and (3.8) would imply that The desired estimate trivially follows from (3.7) if λ 1 > 0. So, we now deal with the other case λ 1 = 0 and apply a well-known trick based on the assumption (2.22) and the consequent inequality which holds for some C 0 > 0 and every s ∈ R and λ ∈ (0, 1), where δ 0 is such that the interval [m 0 − δ 0 , m 0 + δ 0 ] is included in the interior of D(β) (cf. [ and this can be used when testing equation (3.3) by ϕ λ σ − m 0 1. To this concern, we recall that 1 ∈ V σ B by (2.10) and notice that the conservation property (2.31) also holds for ϕ λ σ , i.e., mean ϕ λ σ (t) = m 0 for every t ∈ [0, T ]. So, for a.a. t ∈ (0, T ), we test (3.3) by ϕ λ σ (t) − m 0 1 and rearrange a little. However, we omit writing the time t for a while. We also write k instead of k1 if k is a real number. We have a.e. in (0, T ) that The left-hand side of this equality can be estimated from below by virtue of (3.12). The first term on the right-hand side can be dealt with by accounting for the Poincaré type inequality (2.13) as follows: the last equality being due to A r 1 = 0. Therefore, by recalling (3.7), we have that the whole right-hand side of (3.13) is bounded in L 2 (0, T ) and conclude that At this point, we can test the second equation (3.3) by 1 and deduce a bound for mean µ λ σ in L 2 (0, T ). This and (3.7) imply (3.9). As already noticed, (3.10) is proved as well.
First conclusion. As already remarked, in the proof of [15, Thm. 5.1] with a fixed σ it is shown that (ϕ λ σ , µ λ σ ) converges as λ tends to zero (in a proper topology, possibly along a subsequence) to some pair (ϕ σ , µ σ ), and it is proved that such a pair is a solution to problem (2.27)-(2.29). We prove that the family {(ϕ σ , µ σ )} σ>0 constructed in this way satisfies all the requirement of the statement. The starting point is the conservation of the bounds just proved in the limit as λ ց 0. We have that ϕ σ H 1 (0,T ;H) + µ σ L 2 (0,T ;V 2r A ) + B σ ϕ σ L ∞ (0,T ;H) ≤ c , and we conclude that (2.33)-(2.35) hold true for some triplet (ϕ, µ, ζ) satisfying (2.36). This ends the proof of the first part of the statement.
First characterization. We are going to show that ζ = ϕ − P ϕ by proving that B σ ϕ σ → ϕ − P ϕ weakly in L 2 (Q). (3.14) To this end, we use the eigenvalues λ ′ j and the eigenfunctions e ′ j of B and notice that e ′ j is orthogonal to ker B if λ ′ j > 0 while λ ′ j = 0 if e ′ j ∈ ker B. We set, for convenience, for ψ ∈ L 2 (0, T ) and j = 1, 2, . . . , and we notice that (3.14) follows if we prove that for every ψ and j as before, since the linear combinations of the products ψ e ′ j of such real functions and eigenfunctions of B form a dense subspace of L 2 (Q). So, we fix ψ and j. As for j, we distinguish two cases. Assume first that λ ′ j > 0. Then, (λ ′ j ) σ tends to 1 as σ tends to zero. Moreover, (2.33) holds. We thus deduce that the last equality being due to the orthogonality between P ϕ(t) and e ′ j . Assume now that λ j = 0. Then, we trivially have that A σ j (ψ) = 0 for every σ > 0. On the other hand, we also have that A 0 j (ψ) = 0 since e ′ j ∈ ker B and ϕ(t) − P ϕ(t) is orthogonal to ker B for a.a. t ∈ (0, T ). Therefore, (3.15) is proved in any case.
To conclude the proof, we have to show that (ϕ, µ) solves problem (2.38)-(2.40) under the further assumption (2.37). The first equation obviously follows from (2.27) due to (2.33)-(2.34), and the initial condition (2.40) is satisfied as well since (2.33) implies weak convergence in C 0 ([0, T ]; H). So, it remains to verify the variational inequality (2.39). To this concern, it is convenient to give different formulations of both (2.28) and (2.39). This procedure is based on the lemma stated below, which follows from the classical theory of variational inequalities of elliptic type in the framework of Convex Analysis. However, for the reader's convenience, we also give a simple proof. Lemma 3.2. Let V be a Hilbert space, V * its dual space, · , · the duality pairing between V * and V , and a : V × V → R a continuous bilinear form. Moreover, assume that γ 1 : V → (−∞, +∞] is convex, proper and lower semicontinuous, (3.17) γ 2 : V → R is convex and Gâteaux differentiable, and γ 2 : V → V * is its Gâteaux derivative. Then, for every u ∈ V and g ∈ V * , the variational inequalities (3.20) are equivalent to each other.
Proof. Assume (3.19) and let v ∈ V . Since γ 2 is convex and γ 2 is its derivative, we have that follows, that is, (3.20). Assume now (3.20) and let v ∈ V . By writing (3.20) with w in place of v and then choosing By rearranging and dividing by ϑ, we deduce that On the other hand, the convexity of γ 1 implies that By combining these inequalities, we deduce that and letting ϑ tend to zero, we obtain (3.19).
As already announced, we use this lemma to replace both (2.28) and (2.39) by different variational inequalities.
First alternative formulation. We first observe that (2.28) for every v ∈ V σ B as required implies the same inequality for every v ∈ V σ 0 B since V σ 0 B ⊂ V σ B . Now, by recalling that L π is the Lipschitz constant of π, we replace the latter variational inequality by an equivalent one by applying lemma with the choices Notice that γ 2 actually is convex (since π ′ + L π ≥ 0 a.e. in R) and Gâteaux differentiable and that its derivative γ 2 is given by γ 2 (u), v = (π(u) + L π u, v). Hence, we deduce that the variational inequality (2.28) required just for every v ∈ V σ 0 B is equivalent to for every v ∈ V σ 0 B and for a.a. t ∈ (0, T ). (3.24) Unfortunately, this does not seem to be true, in general, and we prove the following: the variational inequality (2.39) with v varying in V σ 0 B is equivalent to (3.24). (3.25) To this aim, it suffices to apply the lemma with the same γ i as before and obvious u and g, but with V = V σ 0 B and a defined by a(u, v) : Conclusion of the proof. In view of (3.23) and (3.25), our aim is first to verify (3.24) by starting from (3.21) (implied by (2.28)), while (2.39), as it is, will be proved at the end by accounting for (2.37). However, the left-hand side of (3.21) contains the quadratic term associated to the map v → −L π Ω |v| 2 . This term is unpleasant since the related map is concave. To get rid of it, we adapt the procedure introduced in [13] to the present case. We set, for convenience, κ := L π τ , ρ σ (t) := e −κt ϕ σ (t) and ρ(t) := e −κt ϕ(t) for a.a. t ∈ (0, T ), (3.26) and we notice that w → Q e −2κt w 2 is the square of an equivalent norm in L 2 (Q). At this point, we pick an arbitrary v ∈ L 2 (0, T ; V σ 0 B ), write (3.21) by taking v(t) as test function, multiply by e −2κt , and integrate over (0, T ). We obtain (3.27) Well, we want to take the limit as σ tends to zero in this inequality. As for the first term on the left-hand side, we have that By observing that ρ σ converges to ρ weakly in H 1 (0, T ; H), thus weakly in C 0 ([0, T ]; H), so that ρ σ (T ) converges to ρ(T ) weakly in H, we therefore have that Next, by (3.14) and the lower semicontinuity of the norms, we have that By also recalling (3.16), we can write By taking the difference, we deduce that the last equality being due to the othogonality between (ϕ − P ϕ)(t) ∈ (ker B) ⊥ and (P ϕ − P v)(t) ∈ ker B. Moreover, by observing that the functional w → Q e −2κt α(w) is lower semicontinuous on L 2 (Q), and recalling that ϕ σ converges to ϕ weakly in L 2 (Q), we deduce that This ends the treatment of the terms on the left-hand side of (3.27). Concerning the righthand side, we have to overcome the difficulty due to the coupling between µ σ and ϕ σ . To this end, we introduce the notation and deduce from (2.30) that Hence, we have that Now, from (2.34) we deduce that 1 * µ σ converges to 1 * µ weakly in is compact, we infer that 1 * µ σ converges to 1 * µ strongly in L 2 (0, T ; V r A ). In view of (2.38) and (2.40), we deduce that Finally, by recalling (2.32), we see that the term involving f σ and the last one of (3.27) do not give any trouble. Therefore, we conclude that (3.28) and this holds for every v ∈ L 2 (0, T ; V σ 0 B ). On the other hand, (3.28) is equivalent to τ e −κt ∂ t ϕ(t) − κϕ(t) , e −κt (ϕ(t) − v)(t) for a.a. t ∈ (0, T ) and every v ∈ V σ 0 B . By multiplying by e 2κt and recalling that κ = L π /τ , we obtain (3.24) as claimed. Recalling (3.25), we have proved that the variational inequality (2.39) is satisfied for every test function v ∈ V σ 0 B . At this point, we account for (2.37), not yet used up to now, and show that (2.39) actually holds for every v ∈ H. To this end, for a given v ∈ H with β(v) ∈ L 1 (Ω) without loss of generality, it suffices to take a sequence {v n } given by (2.37), test (3.24) by v n and let n tend to infinity. One obtains (3.24) for v without any trouble. This completes the proof. Remark 3.3. Going back to the above proof, one justifies what has been announced in Remark 2.7: if (2.37) is not assumed, one anyway arrives at the variational inequality (2.39) required for every v ∈ V σ 0 B instead of for every v ∈ H. Indeed, (2.37) has been only used at the end, in order to extend to any v ∈ H the validity of (2.39) already proved for test functions v ∈ V σ 0 B .

The limiting problem
In this section, we prove Theorem 2.10 and Theorem 2.11. As far as the former is concerned, some preliminaries are needed. We refer to [15,Sect. 3] for more details. We set V −r A := V r A * and · A,−r := the dual norm of · A,r , and we use the symbol · , · A,r for the duality pairing between V −r A and V r A . It is understood that H is identified with a subspace of V −r A in the usual way, i.e., in order that v, w A,r = (v, w) for every v ∈ H and w ∈ V r A . Moreover, we introduce the subspaces V ±r and that A 2r 0 is an isomorphism between V r 0 and V −r 0 . Thus, we can set A −2r 0 := (A 2r 0 ) −1 and obtain an isomorphism between V −r 0 and V r 0 . It also turns out that Finally, the following formula holds true: a.e. in (0, T ), for every ψ ∈ H 1 (0, T ; V −r 0 ).
In particular, Proof of Theorem 2.10. We just prove the continuous dependence part, since uniqueness for the first component follows as a consequence. We set, for convenience, f := f 1 −f 2 , ϕ := ϕ 1 − ϕ 2 , and µ := µ 1 − µ 2 . Now, we write equation (2.38) at the time s for these solutions and take the difference. Then, we test the resulting identity by v = A −2r 0 ϕ(s), where we observe that ϕ(s) ∈ V −r 0 , since ϕ ∈ C 0 ([0, T ]; H) by (2.36) and mean ϕ(s) = 0 if λ 1 = 0 by the conservation property (2.31), so that v is a well-defined element of V r A . Moreover, we have that A −2r 0 ϕ ∈ L ∞ (0, T ; V r A ). Integrating over (0, t) with respect to s, where t ∈ (0, T ) is arbitrary, we obtain the identity Now, the first term on the left-hand side is nonnegative by (4.2). Hence, by also noting that µ ∈ L 2 (0, T ; V r A ) and applying (4.1), we deduce that At the same time, we write (2.39) for f i and (ϕ i , µ i ), i = 1, 2, test them by ϕ 2 and ϕ 1 , respectively, add the resulting inequalities to each other, and integrate over (0, t) as before. Then, the terms involving β cancel out. By denoting by I the identity map of H and rearranging, we have that We observe that I − P is the projection operator on the orthogonal subspace (ker B) ⊥ . It follows that ((I − P )v, v) = ((I − P )v, (I − P )v) ≥ 0 for every v ∈ H, so that the second term on the left-hand side of (4.4) is nonnegative. By adding (4.3) and (4.4) to each other, and accounting for this observation, an obvious cancellation, the Lipschitz continuity of π and the Schwarz and Young inequalities, we deduce that By applying the Gronwall lemma, we conclude that the desired estimate (2.48) holds true with a constant C cd as in the statement.
Finally, we prove Theorem 2.11. The proof we give is based on the study of the auxiliary problem of finding φ ∈ H 1 (0, T ; H) satisfying We have subtracted the constant π(0) to π(φ(t)) in (4.5) in order to use the inequality |π(s) − π(0)| ≤ L π |s| for s ∈ R without any additive constant. This is needed in the sequel, indeed. Since β is convex, P is linear and π is Lipschitz continuous, this problem has a unique solution φ provided that the initial datum also satisfies β(φ 0 ) ∈ L 1 (Ω) .
for some positive constants C 1 and C 2 . Then, for every ϑ ∈ (0, 1) and p ∈ [1, +∞], we have that with a constant C that does not depend on T.
In the above lemma, the symbol · X stands for the norm in the generic Banach space X. The same convention is followed in the rest of the section, where · X also denotes the norm in the power X 3 (however, we keep the short notation · without indices if X = H). Moreover, (X, Y ) ϑ,p is the real interpolation space between the Banach spaces X and Y (for basic definitions and properties see, e.g., [27,Sect. 1.1]).
Lemma 4.2. Let the general assumption on the structure be fulfilled and assume that the data g and φ 0 satisfy g ∈ L 2 (0, T ; H η (Ω)) and φ 0 ∈ H η (Ω) (4.12) for some η ∈ (0, 1], as well as (4.8). Then, the solution φ to problem (4.5)-(4.6) enjoys the further regularity φ ∈ L 2 (0, T ; H η (Ω)), (4.13) and there exists some ξ satisfying ξ ∈ L 2 (0, T ; H) and ξ ∈ β(φ) a.e. in Q , (4.14) (4.15) Proof. By still denoting by β λ and β λ the Moreau-Yosida approximations of β and β, respectively, we introduce the approximating problem of finding φ λ ∈ H 1 (0, T ; H) that satisfies τ ∂ t φ λ + φ λ − P φ λ + β λ (φ λ ) + π(φ λ ) − π(0) = g a.e. in Q (4. 16) and the initial condition (4.6). For any data satisfying (4.7) (while (4.8) is not needed here) also this problem has a unique solution φ λ . We perform some a priori estimates. As usual, the symbol c stands for possibly different constants. In this proof, the values of c can only depend on τ , π, Ω, T and the eigenfunctions e ′ j associtated to the zero eigenvalues of B (if any). In particular, they do not depend on λ, nor on the data of problem (4.5)-(4.6). Symbols like C and C i denote particular values of c we want to refer to. The first three estimates we perform are in the direction of the inequalities (4.9) and (4.10) which we want to satisfy with a suitable choice of the spaces and the operator. For this reason, they are obtained under different regularity assumptions on the data.
First a priori estimate. Let g i and φ 0,i , i = 1, 2, be two choices of the data satisfying (4.7) and let φ λ,i be the corresponding solutions to the approximating problem. We set for brevity φ λ := φ λ,1 − φ λ,2 , g := g 1 − g 2 and φ 0 := φ 0,1 − φ 0,2 . We write (4.16) for both solutions, take the difference and multiply it by φ λ . Then, we integrate over Q t . We obtain that Since β λ is monotone, all of the terms on the left-hand side are nonnegative. By estimating the right-hand side on account of the Lipschitz continuity of π and the Schwarz and Young inequalities, and then applying the Gronwall lemma, we easily conclude that It trivially follows that Second a priori estimate. We assume (4.7) on the data. By multiplying (4.16) by φ λ and integrating over Q t , we obtain that All of the terms on the left-hand side are nonnegative since β λ is monotone and β λ (0) = 0.
If we estimate the right-hand side by using the Lipschitz continuity of π and the Schwarz and Young inequality, we immediately deduce that Third a priori estimate. We set V := H 1 (Ω) for brevity and assume that the data satisfy g ∈ L 2 (0, T ; V ) and φ 0 ∈ V . Before going on, we make an observation. Assume first that ker B = {0}. Then P = 0 and (4.5) is an ordinary differential equation where the space variable is just a parameter. In the opposite case, the presence of the nonlocal operator P could be unpleasant. However, we are reduced to the same situations as before by moving the term P φ λ to the right-hand side and treating it as a datum. More precisely, in this case, ker B has a finite dimension m > 0 and is spanned by the first m eigenfunctions (those corresponding to the zero eigenvalues). Since every eigenfunction of B belongs to the domain V n B of B n for every n ∈ N and we are assuming (2.49), the eigenfunctions (we are interested in) belong to V , and we have the identities Namely, we have that P v ∈ V even though v only belongs to H. Therefore, in any case, the solution φ λ enjoys some space regularity. Precisely, it belongs to L 2 (0, T ; V ) as well as its time derivative and we have that e. in Q . By multiplying this equation by ∇φ λ and integrating over Q t , we obtain that All of the terms on the left-hand side are nonnegative. The volume integrals on the righthand side, except the one involving P , can be easily treated thanks to the boundedness of π ′ and the Schwarz and Young inequalities. If P = 0, then we can apply the Gronwall lemma and obtain an estimate of ∇φ λ . Recalling (4.19), we conclude that We claim that the same estimate holds true even though ker B is nontrivial. In this case, we recall the representation formula (4.20) and apply it to φ λ . By also accounting for standard inequalities, we obtain that So, it suffices to recall (4.19) and apply the Gronwall lemma to obtain (4.21) also in this case. Therefore, (4.21) is established and it trivially implies that φ λ L 2 (0,T ;V ) ≤ C 2 ( g L 2 (0,T ;V ) + φ 0 V ) . Interpolation. Now, let the data satisfy (4.12) with η ∈ (0, 1). We choose and apply Lemma 4.1 to the operator T : A 1 → B 1 that associates to the pair (g, φ 0 ) the solution φ λ to problem (4.5)-(4.6). Then, (4.18) and (4.22) yield (4.9) and (4.10), respectively. Moreover, by setting ϑ := 1 − η, we have that so that (g, φ 0 ) ∈ (A 0 , A 1 ) ϑ,2 by (4.12). It follows that φ λ ∈ (B 0 , B 1 ) ϑ,2 = L 2 (0, T ; H η (Ω)) and φ λ L 2 (0,T ;H η (Ω)) ≤ CC ϑ 1 C 1−ϑ 2 (g, φ 0 ) L 2 (0,T ;H η (Ω))×H η (Ω) (4.23) with a constant C that does not depend on λ. Notice that (4.23) with η = 1 (i.e., ϑ = 0) is ensured by (4.22).
Fourth a priori estimate. We are close to the conclusion, and we thus assume that the data g and φ 0 are as in the statement. By multiplying (4.16) by ∂ t φ λ , integrating over Q t , and rearranging, we have that Since β λ is nonnegative and β λ (φ 0 ) ≤ β(φ 0 ) a.e. in Ω, owing to the Schwarz and Young inequalities and the Lipschitz continuity of π, and accounting for (4.8) and (4.19), we infer that A comparison in (4.16) then yields that possibly only for a subsequence λ k ց 0. Then, φ(0) = φ 0 , and (4.15) is verified. Moreover, by also applying, e.g., [4, Lemma 2.3, p. 38], we infer that (φ, ξ) satisfies the inclusion in (4.14). On the other hand, all this implies (4.5) since β is convex, so that φ is the solution to problem (4.5)-(4.6). This completes the proof of the lemma.
At the end of the proof of Theorem 2.11, we arrive at the regularity ϕ ∈ (L ∞ (0, T ; V ), L ∞ (0, T ; H)) ϑ,∞ (4.28) for the first component ϕ of the solution (ϕ, µ) to problem (2.38)-(2.40). We avoid the troubles that may arise with the exponent ∞ and do not offer a different representation of the space appearing in (4.28). We just remark that the regularity (4.28) is neither better nor worse than (2.41), since it yields some better time regularity at the expense of a lower space regularity. One can prove that (L ∞ (0, T ; V ), L ∞ (0, T ; H)) ϑ,∞ ⊂ L ∞ (0, T ; H η−ε (Ω)) for every ε > 0 (in particular, the Aubin-Lions lemma can be applied also in the modified proof of Lemma 4.2) so that the Sobolev type regularity for ϕ we can obtain is ϕ ∈ L ∞ (0, T ; H η−ε (Ω)) for every ε > 0 . Let us give some detail. Assume that (ϕ, µ i ), i = 1, 2, are solutions corresponding to some data ϕ 0 and f (with the same first component, due to Theorem 2.10). By writing (2.38) for both solutions and taking the difference, we immediately obtain that (A r (µ 1 − µ 2 ), v) = 0 for every v ∈ V r A and a.e. in (0, T ), that is This implies that µ 1 = µ 2 if λ 1 > 0. In the opposite case λ 1 = 0, we can arrive at the same conclusion under additional conditions, as we show at once by following the ideas of [15,Rem. 4.1]. However, in the present case, the condition we assume on the solutions is difficult to verify, unfortunately. Suppose that D(β) is an open interval, the restriction of β to D(β) is a C 1 function, and all of the values attained by ϕ belong to a compact subinterval [a, b] ⊂ D(β). Now, choose δ 0 such that the interval [a − δ 0 , b + δ 0 ] is contained in D(β). Then, for an arbitrary δ ∈ (0, δ 0 ) and for a.a. t ∈ (0, T ), we can choose v = ϕ(t) − δ (whence ϕ(t) − v = δ) and v = ϕ(t) + δ (whence ϕ(t) − v = −δ) in the variational inequality (2.39) written for (ϕ, µ 1 ) and (ϕ, µ 2 ), respectively. Then, by adding the resulting inequalities, we deduce that δ) a.e. in (0, T ).