LONG TIME BEHAVIOR OF THE CAGINALP SYSTEM WITH SINGULAR POTENTIALS AND DYNAMIC BOUNDARY CONDITIONS

This paper is devoted to the study of the well-posedness and the long time behavior of the Caginalp phase-field model with singular potentials and dynamic boundary conditions. Thanks to a suitable definition of solutions, coinciding with the strong ones under proper assumptions on the bulk and surface potentials, we are able to get dissipative estimates, leading to the existence of the global attractor with finite fractal dimension, as well as of an exponential attractor.

1. Introduction. The Caginalp system is a well-known model in phase transition, proposed in [1] to describe, in particular, melting-solidification phenomena in certain classes of materials. It consists of two parabolic equations in the state variables (w, u), the temperature and the order parameter, respectively. Here, we assume that the material undergoing phase transition is contained in a vessel, so that interactions with the walls need to be considered. For this purpose, physicists proposed, in the context of the Cahn-Hilliard equation, the so-called dynamic boundary conditions for the order parameter (in the sense that the kinetics, i.e., the time derivative of the order parameter, appears explicitly in the boundary conditions), see [7], [8] and [9].
The mathematical study of the Cahn-Hilliard model with singular potentials F (here and below, the nonlinear term f is the derivative of the potential F ) and/or dynamic boundary conditions has been developed in many papers, see, e.g., [12], [13], [16], [17] and [19]. The Caginalp model with singular potentials and dynamic boundary conditions has been considered for the first time in [4], assuming that the nonlinearity on the boundary has the right signs close to ±1. With this requirement, the first and third authors were able to prove the well-posedness (and, in particular, the separation from the singularities), together with the dissipativity of the system, the existence of the global attractor and the convergence of solutions to steady states (this last issue when the nonlinear term on the boundary disappears and the potential is real analytic in (−1, 1)). Unfortunately, the sign restrictions on g exclude, e.g., constant functions and, in [2], the authors tried to remove this assumption, with the result that even the well-posedness of the problem becomes a difficult task. Indeed, only the global existence (and uniqueness) of strong solutions (always separated from the singularities of f ) was proved, provided that the singularities of the potential are strong enough, relying on a suitable elliptic problem, as well as on localization techniques, since reasonable super/sub-solutions were not available. This did not prevent the solutions from blowing up as the initial data approach the singularities, so that the asymptotic analysis was not possible, unless some technical assumptions related to the terms appearing in the dynamic boundary conditions are imposed (see [3]). In all these papers, strong solutions were considered, whereas, as we will see, the occurrence of both singular potentials and nonlinear dynamic boundary conditions gives rise to complicated dynamics.
A way to overcome such difficulties consists in using duality techniques, as in [14], where a problem similar to ours, with dynamic boundary conditions for the temperature as well, has been addressed (see also [20] for a Cahn-Hilliard model). Again, the existence of attractors is proved under sign or growth restrictions on the nonlinear terms.
An analogous outcome, in the context of the Cahn-Hilliard equation, was deeply analyzed by Miranville and Zelik in [19], exhibiting the possible appearance of strong singularities close to the boundary, in particular, when the aforementioned sign restrictions are not satisfied, and showing that, already in the 1-D case, solutions in the sense of distributions may not exist, due to the jumps of the normal derivative close to the boundary. Thus, the authors modified the notion of a solution, by introducing a variational inequality (we can note that the aforementioned duality techniques are somehow similar to this approach, but at an abstract level), and proved that such a solution is the usual one when it does not reach the singular values on the boundary, a fact prevented by a fast growing nonlinear term f or by the sign conditions on g.
Our aim in this paper is to extend this approach to the Caginalp system which, as the numerical simulations at the end of the paper point out, in presence of a logarithmic potential and without the sign requirements on g, exhibits solutions stopping at a certain time when they reach one of the singular values ±1 on the boundary. On the contrary, there exist global solutions, provided that g has proper signs at ±1. In other words, it is again relevant to adopt the variational definition of a solution, which allows, in particular, to prove the uniform in time estimates needed for dissipativity, without any additional assumption on f and g. Having gained the lacking ingredient, we are now in a position to accomplish our asymptotic analysis and prove the existence of finite-dimensional attractors.
This paper is organized as follows. Section 2 is devoted to our assumptions and notation. In Section 3, we introduce regularized problems in which the singular nonlinearity is approximated by regular functions, obtaining uniform a priori estimates on the corresponding solutions. Then, a variational formulation of (1) is given, for which well-posedness and regularity estimates are proved in Section 4. In fact, under the sufficient conditions stated in Section 5, we prove that a variational solution coincides with a solution in the usual (distribution) sense. Then, the existence of finite-dimensional (global and exponential) attractors is shown in Section 6. In Section 7, we give numerical simulations which suggest the possible nonexistence of classical solutions. Finally, in Appendix, we recall, for the reader's convenience, some results of [19] which are used in our proofs.

2.
Setting of the problem. We setf (u) := f (u) − λu and rewrite (1) in the form where f is a singular function satisfying As a consequence, the following properties hold forf : whereF (s) = s 0f (r)dr andc,C are strictly positive constants. The nonlinear function g ∈ C 2 ([−1, 1]) can be extended, without loss of generality, to the whole real line by writing

LAURENCE CHERFILS, STEFANIA GATTI, ALAIN MIRANVILLE
In the whole paper, · Γ and ·, · Γ stand for the norm and the scalar product in L 2 (Γ) (or [L 2 (Γ)] 3 , depending on the context), respectively; moreover, we denote by · and ·, · the norm and the scalar product or the duality pairing in L 2 (Ω) (or, again, [L 2 (Ω)] 3 or [L 2 (Ω)] 6 ). Finally, we introduce the spaces both endowed with their standard norms. We then set, concerning the temperature, Besides, for further convenience, given two normed function spaces X in Ω and Y on Γ, we set, whenever it makes sense, and we endow this space with the norm The problem is characterized by the conservation law where v = 1 |Ω| Ω vdx. Thus, we will often use the obvious inequalities and Besides, we introduce the spaces (10) and the cartesian products The symbol ·, · in [H 1 (Ω)] stands for the duality pairing between H 1 (Ω) and [H 1 (Ω)] * . Notice that [H 1 (Ω)] is not the dual space ofH 1 (Ω). In particular, we denote by c Ω the positive constant such that We further use −∆ as the realization of the Laplacian with homogeneous Neumann boundary conditions acting on the space of the L 2 (Ω)-functions with null average (it is thus a positive invertible operator with compact inverse (−∆) −1 ).
For the sake of simplicity, throughout the whole paper, c denotes any positive constant, allowed to vary in the same line and independent of the initial data, but possibly influenced by the average I 0 , ε and the other structural parameters (λ, K, etc.). Further dependencies will be specified on occurrence.

3.
A priori estimates for approximating regular problems. Since the interaction between the singular potential and the dynamic boundary condition may give rise to singular solutions, we first focus on approximating problems for which the former difficulty is weakened. Following [19], we introduce a family of regular approximating functions: given any N ∈ N, we set Then, we call F N the primitive F N (s) = s 0 f N (r)dr and, having setf N (s) = f N (s) − λs, we defineF N analogously, withf N in place of f N . For the reader's convenience, we recall a particular instance of [19, formulae (2.14) and (2.17), with c = 0], namely, there exist α > 0 and c > 0 such that, for N large enough, there hold together with f N (s)s ≥ F N (s) ≥ 0. Our first aim is to study the family of problems The well-posedness of these problems is already well established (see, e.g., [11]) and will not be considered in this paper. Unfortunately, we cannot find a uniform (with respect to N ) H 2 (Ω)-estimate on u N . Nevertheless, we can control some Hölder norm of u N in Ω, together with the H 2 -norm in some interior domain and the L 2boundary norm of the gradient of the tangential derivative, ∇D τ u N (we recall that D τ u = ∇u − (∂ n u)n is the tangential part of the gradient ∇u and n(x) stands also for some smooth extension of the unit normal vector field at the boundary inside Ω). As it will be clear below, this is enough for our purpose.
For the sake of simplicity, we drop the subscript N in u N , ψ N , w N . In particular, we recall that the positive constants c appearing below are independent of N and the initial data.
The main result of this section is the Theorem 3.1. We assume that f and g satisfy the above assumptions. Then, for N large enough, any sufficiently regular solution z(t) = (u(t), ψ(t), w(t)) to (14) is such that u ∈ C α (Ω) ⊗ H 2 (Γ), for some α < 1/4, F N (u) ∈ L 1 (Γ), ∇D τ u ∈ [L 2 (Ω)] 6 , u ∈ H 2 (Ω δ ), where Ω δ = {x ∈ Ω : dist(x, Γ) > δ}, for every δ > 0, and there where the positive constant ν is independent of N and the initial data. Moreover, the following smoothing property holds: Proof. The estimates on w and ∂ t z are obtained in Lemma 3.6 below. In order to control (u, ψ), we follow the rationale of [19], that is, we consider the nonlinear elliptic problem Then, we apply Theorem 8.1 (namely, [19,Theorem 6.1] which is written in Appendix for the reader's convenience): this is possible, provided thath 1 (t) ∈ L 2 (Ω) andh 2 (t) ∈ L 2 (Γ) and that their norms can be (uniformly in N ) controlled by the initial data. Actually, by definition ofh 1 andh 2 , from the assumptions on g, it is straightforward to get . Hence, we are led to establish proper estimates on z(t) L 2 , ∂ t u(t) and ∂ t ψ(t) Γ , a task that we accomplish in the next lemmas.
We are thus left to prove several technical lemmas which are needed in the proof of the above Theorem 3.1.
Lemma 3.2. There exists ν > 0 small enough such that, for any t ≥ 0 and N large enough, we have (18) for some positive constants c which are independent of N , but monotone increasing with respect to |I 0 |, 1/ε and 1/ν, where I 0 = u 0 + εw 0 . Moreover, for any t > 0, there holds LONG TIME BEHAVIOR OF A CAGINALP SYSTEM 7 Proof. Multiplying the second equation of (14) by u in L 2 (Ω) and exploiting decomposition (5), we have Next, we rewrite the first equation of (14) as , ∀t > 0. Then, multiplying the above equation by (−∆) −1 (u + εw − I 0 ), we get This, by (9), reduces to We also take the product of the above equation by ε(u + εw), getting Adding the three equalities, introducing the functional Thus, since g 0 is a globally bounded function, by (9), we have, concerning the right-hand side of the above differential equation, (5) and now c also depends on |I 0 |. Collecting the above estimates and taking N large enough for (13) to hold, we end up with Notice that (9) and (11) entail , which allows to see that, for some c > 1, there holds
For t ≥ 1, N large enough and ν > 0 small enough, we have Proof. Taking the product in L 2 (Ω) of the first equation of (14) first by w, and then by ∂ t w, and adding this second equation multiplied by ε to the first one, we obtain The sum of this equality with the product in L 2 (Ω) of the second equation of (14) by ∂ t u, namely,
Lemma 3.5. We have, for all t ≥ 1 and N large enough, Moreover, for any t > 0 and N large enough, there holds Proof. The first formula is a direct consequence of Lemmas 3.3 and 3.4. Indeed, from (30), we have , which, plugged into (28), allows to conclude. Inequality (31) then follows from this first estimate and (30).
Lemma 3.6. There holds, for all t ≥ 0 and N large enough, Moreover, for t > 0, we have the smoothing property Proof. Having set θ = ∂ t u, ζ = ∂ t ψ, a differentiation of the second and third equations of (14) with respect to time gives Taking the product of the first equation of (34) by θ in L 2 (Ω), we obtain Then, multiplying the first equation of (14) by −ε∆∂ t w in L 2 (Ω), we have Adding the above equations to (24), thanks to (4) and (5), we are led to
We conclude this section with an estimate on the difference of two solutions to problem (14) which furnishes the Lipschitz continuous dependence of the solutions on the initial data and, in particular, the uniqueness of solutions to problem (14).

4.
Variational formulation and well-posedness. This section is devoted to the definition of a suitable notion of a solution to the limit problem, that is, the problem obtained by letting N → +∞ and which formally coincides with (2). The difficulty is that we should allow the solutions to reach the singular values (i.e., the pure phases) on the boundary and, at the same time, get a well-posed problem, whose solutions coincide with the classical ones under proper assumptions (e.g., sign conditions). All the following computations are formal. The first equation of (2) can be rewritten as Taking advantage of this notation and multiplying the second equation of (2) by u − v 2 , for any v 2 ∈ H 1 (Ω) ⊗ H 1 (Γ), we get Then, from (53) and the monotonicity of f , we infer If we consider the solutions to (14) departing from initial data in and then pass to the limit N → +∞, we find functions living in Φ for any time, as we will rigorously see in this section (cf. Theorem 4.3). These functions are not necessarely solutions to (2) in the usual sense and, arguing as in [19,Section 3], we suitably modify the notion of a solution as follows.
We emphasize that we do not assume in the definition that ψ 0 is the trace of u 0 . In order to show the uniqueness of a variational solution, we consider (54)-(55) in terms of test functions v 1 = v 1 (x, t) and v 2 = v 2 (x, t), with v 1 , v 2 satisfying the regularity assumptions in Definition 4.1, and integrate (54)-(55) with respect to t, a legitimate step, since all terms are L 1 in time. This gives, for t > s > 0, Arguing as in [19], the function v α = (1 − α)u + αv 2 , where α ∈ (0, 1] and v 2 is an arbitrary admissible test function, is an admissible test function for (57) as well (indeed, f (v α (t)) ∈ L 1 (Ω) follows from (3) 4 which implies that |f (·)| is convex). Then, taking the corresponding (57), where we recall that u is absolutely continuous on [s, t] with values in [L 2 (Ω)⊗L 2 (Γ)], we simplify by α and pass to the limit α → 0, getting thanks to the Lebesgue dominated convergence theorem.
We can now state the following Lemma 4.2. For every two variational solutions z i (t) = (u i (t), ψ i (t), w i (t)) departing from z i = (u i , ψ i , w i ), i = 1, 2, we have the following estimate on the difference z 1 (t) − z 2 (t) = (ū(t),ψ(t),w(t)) in terms of the initial datum z 1 − z 2 = (ū 0 ,ψ 0 ,w 0 ): Proof. We sum (58), with u = u 2 , w = w 2 , v 2 = u 1 , and the variational inequality This provides, after obvious simplifications, Then, recalling (7), we use (56), with w = w 1 , u = u 1 , v 1 =ū + εw (respectively, with w = w 2 , u = u 2 , v 1 =ū + εw), and find, after subtracting the two resulting equalities and since w ,ū + εw − ū + εw = 0, Adding this equation to (59), we see that the functional and, thanks to the Gronwall lemma, we find for some positive constant c which is independent of s and t. Passing to the limit s → 0 and owing to the continuity of z 1 , z 2 (cf. Definition 4.1), we get the desired estimate on ū(t) 2 + ψ (t) 2 Γ . Arguing exactly as in the proof of Lemma 3.7, we control the w-term and we finally have the estimate of Lemma 4.2, which, in particular, gives the uniqueness of a variational solution.
We now prove the existence of a variational solution in the following Theorem 4.3. For every initial datum z 0 = (u 0 , ψ 0 , w 0 ) ∈ Φ, problem (2) possesses a unique variational solution z(t) = (u(t), ψ(t), w(t)) in the sense of Definition 4.1. Such a solution regularizes as t > 0 and all the uniform estimates obtained above (except for the one on F N (u N (·)) L 1 (Γ) ) hold. More precisely, for every δ > 0 and t > 0, we have Furthermore, for any pair of initial data z 1 , z 2 , the difference of the corresponding solutions z 1 (t) and z 2 (t) satisfies for every admissible test functions v 1 and v 2 and t > s > 0. Our task is to pass to the limit N → +∞.
Proof. Owing to the Hölder continuity of u with respect to both x and t, there exists > 0 such that Applying Proposition 2, the approximate solutions u N to (14) (converging to u) satisfy where the positive constant c is independent of N , but depends on and (x 0 , t 0 ). Then, without loss of generality, we can assume that , for some proper neighborhood V of x 0 . This, together with (71), leads to the first statement. The second one is just a straightforward consequence of the first one.
As in [19], we can show that, provided that f is strongly singular or that proper sign conditions hold for g, any variational solution u to our problem satisfies (72). The proof is identical to the one in [19] and is thus omitted. Proposition 1. We assume that either lim s→±1 F (s) = +∞ or g(−1) < 0 < g (1). Then, (72) holds. Thus, for all T > 0, [∂ n u] ext = [∂ n u] int , almost everywhere in Γ × [T, +∞], and, for t ≥ T , (u, ψ, w), solves (2) in the usual sense. does not satisfy the first assumption above, asF is bounded in that case.
Theorem 5.3. Assume that there exist M > 0 and p > 2 such that for some positive constants κ i , i = 1, 2. Then, any variational solution u is strictly separated from the singularities ±1 in Ω, namely, for any T > 0, there exists δ T ∈ (0, 1) such that The proof is written in Appendix for the reader's convenience, since it essentially goes as the one of [19,Theorem 4.7].
In view of the conservation law (7), we consider, for each M > 0, the following subset of the phase space Φ: is the unique variational solution to (2) departing from z 0 . Furthermore, this semigroup is Lipschitz continuous in the Ψ-topology, Here, the positive constant c depends on M , but not on t.
This lemma is a direct consequence of Theorem 4.3. Moreover, the semigroup S(t) is dissipative since, by Lemma 3.5 (which also holds for the variational solutions), we infer the existence of R 0 = R 0 (M ) > 0 such that B H 1 (R 0 ), i.e., the H 1 -ball centered at zero with radius R 0 , is absorbing in Φ M and compact in the Ψ-topology. In particular, there exists a time t 0 ≥ 1 such that , for any t ≥ t 0 . As a consequence, the set ],H 1 ) ≤ R, for any initial datum in B 0 . Moreover, by definition of B 0 , there holds u| Γ = ψ.

LAURENCE CHERFILS, STEFANIA GATTI, ALAIN MIRANVILLE
Furthermore, θ satisfies, together with its derivatives, for any k ∈ N, where the constants c k only depend on δ and on the structural data of the problem. The existence of such a cut-off function is ensured by the uniform Hölder continuity of u 0 in Ω, giving the strict separation between ∂Ω δ1 and ∂Ω δ2 , for any δ 1 = δ 2 . Here, we dropped z 0 for the sake of brevity. We denote by B Ψ (z 0 , ρ) the ball in the space B 0 , endowed with the metric of Ψ, centered at z 0 with radius ρ > 0. We then define K z0 as the operator where (u(·), ψ(·), w(·)) is the variational solution departing from z.
Since the first equation of (81) is linear, arguing exactly as in the proof of Lemma 3.2, we obtain Next, the product of the second and the third equations byū andψ, respectively, leads to d dt Thus, summing the last two equations and introducing the energy functional Thanks to (80) and (3) 3 , by definition of θ, In order to suitably control the first term on the right-hand side of the above differential equality, we exploit (8), whence The third term in the right-hand side can be controlled, thanks to the assumptions on g 0 and by the trace interpolation inequality, namely, where C 0 is the constant defined in (5), since we can take Λ large enough to ensure that Λ − λ > 0 and, actually, as large as we want. Replacing these computations in the differential inequality, we obtain d dt Moreover, there holds, for some ω ∈ (0, 2), By (11), we have the control . Thus, taking β > 0 small enough and possibly reducing δ, we can require that Then, simplifying the last differential inequality, we are led to d dt W + βW ≤ c( ū + εw 2 + θū 2 ).
Applying the Gronwall lemma, we finally obtain which yields the thesis, by definition of W .
Proof of Theorem 6.3. Having fixed an arbitrary z 0 ∈ B 0 , for δ, T > 0 and ρ 0 > 0 as in Lemma 6.4, we introduce the spaces Thus, we prove that (S(t), B 0 ) admits an exponential attractor E(M ) by exploiting the -trajectories method [15] (see also [19]). Roughly speaking, we aim to show that, for any ρ ∈ (0, ρ 0 ), the difference of solutions departing from B Ψ (z 0 , ρ) can be decomposed into the sum of a contraction and a smoothing map.
According to Lemma 6.4, for any ρ ∈ (0, ρ 0 ), for any z 1 , z 2 ∈ B Ψ (z 0 , ρ), there holds for some γ ∈ (0, 1). Here, by (62) and Lemma 6.5, the map K z0 satisfies Thus, exploiting the -trajectories method as in [19], we deduce the existence of a (Φ M , Ψ)−exponential attractor E(M ), with basin of attraction B 0 . In particular, . Thus, as in [19], by interpolation (between Ψ and C α (Ω) × C α (Γ) × C α (Ω)), we see that E(M ) has finite dimension and exponentially attracts the bounded sets of Φ M in its natural topology L ∞ (Ω) × L ∞ (Γ) × L ∞ (Ω). Actually, due to (76) and the properties of B 0 , the transitivity of exponential attraction devised in [6] applies, so that the basin of attraction extends to the whole phase space. 7. Numerical results. As far as the numerical simulations are concerned, we use a P 1 finite element approach for the space discretization, together with a semiimplicit Euler time discretization (i.e., implicit for the linear terms and explicit for the nonlinear ones). The numerical simulations are performed with the software Freefem++ [10]. In the numerical results presented below, Ω is a (0, 10) × (0, 4)-rectangle and (2) is endowed with periodic boundary conditions for u and w in the first direction and Neumann for w and dynamic for u boundary conditions in the second one. We further take ε = 0.3,f (u) = −3u + ln( 1+u 1−u ) and g affine. Finally, the initial value consists of uniformly distributed random fluctuations of amplitude ±0.5.
The first two pictures below represent the isovalues of the solution u for different choices of the function g. In the first figure, we take g(u) = u − 0.8, so that the sign conditions are satisfied. As proved in section 5, the solution u stays away from the singularities ±1, for every time, and is thus a classical solution. At time t = 20, u and w have almost converged to a steady state (for w, the steady state is a constant). In Figures 2 and 3, the sign conditions are not satisfied. However, the solution still is classical in Figure 2, where g(u) = u − 1.5. On the contrary, in Figure 3a, where g(u) = u − 3, u reaches the singular value 1 on the boundaries corresponding to dynamic boundary conditions at time t = 0.82 and the simulation stops. Figure 3.b represents the isovalues of w at time t = 0.82. This shows that boundary singularities can appear and suggests nonexistence of classical solutions. Similar simulations were performed in [5] to illustrate nonexistence of classical solutions for the Cahn-Hilliard equation with singular potentials and dynamic boundary conditions.
For the sake of clarity, we recall [19, Proposition 4.1].
Then, u(T ) ∈ H 2 (Ω (T )) and there holds u(T ) H 2 (Ω (T )) ≤ Q ,T , where the positive constant Q ,T only depends on and T , but is independent of the concrete choice of the solution.
The main ingredients of the proof are the Hölder continuity of u and a localization technique, together with regularity results for linear elliptic problems.

LAURENCE CHERFILS, STEFANIA GATTI, ALAIN MIRANVILLE
We now give the proof of Theorem 5.3 which relies on the following lemma, whose rationale is slightly different from the one of [19,Lemma 4.8].
Proof. We rewrite the second and third equations of the problem as and n > 1 is an arbitrary integer, and we integrate over Ω. After simplifications, see [19,Lemma 4.8], we obtain d dt [ ϕ(u) n L n (Ω)⊗L n (Γ) ] + κ ϕ(u) n+2 L n+2 (Ω) + κ ϕ(u) n+1 L n+1 (Γ) ≤ C T,n , where C T,n is independent of time, due to the uniform bounds on h 1 and h 2 , but, of course, depends on n and T . The product by (t − T ) n+1 , followed by an integration over (T, T + 2), gives T +2 T (t − T ) n+1 ϕ(u(t)) n+2 L n+2 (Ω) dt ≤ C T,n , which, thanks to the second inequality of (73) and the arbitrariness of n, allows to conclude.