LONG TIME BEHAVIOR OF AN ALLEN-CAHN TYPE EQUATION WITH A SINGULAR POTENTIAL AND DYNAMIC BOUNDARY CONDITIONS

The aim of this paper is to study the well-posedness and the long time behavior of solutions for an equation of Allen-Cahn type owing to proper approximations of the singular potential and a suitable definition of solutions. We also prove the existence of the finite dimensional global attractor as well as exponential attractors.


Introduction
In this article we are interested in the study of the following initial and boundary value problem, considered in a smooth and bounded domain Ω ⊂ R 3 with boundary ∂Ω = Γ: where λ ∈ R, ∆ Γ is the Laplace-Beltrami operator on the boundary ∂Ω, f and g are given nonlinear interaction functions and λ is some given positive constant.
In particular, f is the derivative of a double-well potential whose wells correspond to the phases of the material. A thermodynamically relevant function f is the following logarithmic (singular) function: The boundary condition will be interpreted as an additional second-order parabolic equation on the boundary ∂Ω. Equation (1.1) may be viewed as a combination of the well-known Cahn-Hilliard equation ∂ t u = −∆(∆u + f (u)), u(0, x) = u 0 (x), and of the Allen-Cahn equation This equation is associated with multiple microscopic mechanisms such as surface diffusion and absorption/desorption and was recently derived and studied in Karali & Katsoulakis [6], Katsoulakis & Vlachos [7], Israel [9], Hildebrand & Mikhailov [10]. This paper is organized as follows. In Section 2, we introduce regularized problems in which the singular nonlinearity is approximated by regular functions and we derive uniform a priori estimates on the corresponding solutions. In Section 3, we formulate the variational formulation of (2.1), we verify the existence and uniqueness of a solution and we study the further regularity of the solutions. In Section 4, we give sufficient conditions which ensure that solutions are separated from the singularities of f and that a variational solution coincides with a solution in the usual (distribution) sense. Finally, we study in Section 5 the asymptotic behavior of the system and we prove the existence of finite-dimensional (both global and exponential) attractors.
We then consider the approximate problems: (2.8) It is convenient to rewrite problem (2.8) in an equivalent form by using the inverse of A := (−∆ + I) (endowed with Neumann boundary conditions). Applying A −1 to both side of (2.8), we obtain: (2.9) We start with the usual energy equality.
Lemma 2.1. Let the above assumptions hold and let ϕ be a sufficiently regular solution of (2.9). Then, the following identities hold: ) , ) ds

11)
and , for some positive constants C and ν which are independent of t.
Proof. Multiplying (2.9) by ϕ and using the fact that: we find: (2.14) Using (2.3), the fact that g 0 is globally bounded and that ∥ψ(t)∥ L ∞ (Γ) ≤ 1, we obtain: d dt for some positive constant C. Integrating (2.15) with respect to t, we deduce: ) , for some constant C. Now, using (2.6) and the fact that g 0 is globally bounded, (2.14) gives: for some positive constants α and C. Hence, for ν > 0 small enough, we obtain: Applying Gronwall's lemma, we deduce estimate (2.11). Finally, integrating (2.17) with respect to t over (t, t + 1) and using (2.11), we obtain: where the constant C is independent of t and of the initial data.

Lemma 2.3.
Let the assumptions of Lemma 2.1 hold, ϕ be a sufficiently regular solution of (2.9) and N be large enough. Then, for t > 0, the following smoothing property holds: ) , where the constant C is independent of N .
Let the nonlinearities f and g satisfy (2.2) and (2.4) respectively and set Ω δ := {x ∈ Ω, d(x, Γ) > δ} . Denote by n = n(x) some smooth extension of the unit normal vector field at the boundary inside the domain Ω. Let also D τ ϕ := ∇ x ϕ − (∂ n ϕ) n be the tangential part of the gradient ∇ x ϕ. Then, for every δ > 0, the following estimate is valid:
Proof. We consider the nonlinear elliptic problem: for every fixed t. Note here that the estimates derived above yield the following control of the right-hand side of (2.49): where C is a positive constant that is independent of N. Due to estimate (2.33), we find that h 1 ∈ L 2 (Ω) and h 2 ∈ L 2 (Γ). Using estimates (2.20), (2.32) and (3.38), we obtain: In order to prove the following estimate: Then, we multiply equation (2.49) by , and we integrate by parts. Using estimate (2.51) and the fact that f ′ ≥ 0, we obtain estimate (2.52). In order to prove: we study the function ϕ in a small ϵ-neighborhood of the boundary Γ. To do so, let x 0 ∈ Γ and y = y(x) be a local coordinates in the neighborhood of x 0 such that y(x 0 ) = 0 and Ω is defined, in these coordinates, by the condition y 1 > 0. Then, we rewrite problem (2.49) in the variable y and after several transformations we find estimate (2.53). To finish the proof of the theorem, we use the following embedding: and we deduce the estimate: Hence, Theorem (2.1) is proved (for more details see Miranville & Zelik [12]).

55)
where the constants C and K are independent of t, N and the initial data.
. Then, this function satisfies the system: Multiplying the first equation of (2.31) by A −1 ϕ, the second equation by ϕ and the third one by ψ and taking the sum of the equations that we obtain, we have the following identity: (2.57) Using (2.3), (2.13) and the fact that g ′ 0 is globally bounded, we obtain: for some positive constants α ′ and C which are independent of N. Using the interpolation inequality ∥u∥ 2 2 ≤ C∥u∥ H 1 (Ω) ∥u∥ H −1 (Ω) and applying the Gronwall inequality, we deduce (2.55).

Variational formulation and well-posedness
This section is devoted to the definition of a suitable notion for a solution to the limit problem, that is, the problem obtained by letting N → +∞ and which coincides with (2.1).To this end, we first fix a constant L > 0 such that: for all φ ∈ H 1 (Ω) and introduce the quadratic form: Then, obviously, we have: The limit problem (2.9), corresponding to N = +∞ formally reads: Multiplying the first equation of (3.4) by the function ϕ − φ, where φ = φ(t, x) is smooth, and integrating by parts, we obtain: which yields: Finally, since B is positive and f is monotone, we have: Consequently, (3.6) can be written as follow: If we consider the solutions of problem (2.9) with initial data belonging to: and then pass to the limit N → ∞, we will find functions living in Φ for all time. These functions are not necessarily solutions to (2.1) in the usual sense. For this, we define a variational solution of the limit problem (3.4) as follows.
We emphasize that we do not assume in the definition that ψ 0 is the trace of ϕ 0 . In order to show the uniqueness of a variational solution, we consider (3.10) in terms of test functions φ = φ(t, x) depending on t and x with φ satisfying the regularity assumptions in Definition 3.1. Then, we write inequality (3.10) with φ = φ(t, x) for almost all t > 0. Moreover, due to the regularity assumptions (3.1) on ϕ and φ, we integrate (3.10) with respect to t since all terms are in L 1 . This gives, for all t > s > 0: (3.11) Arguing as in Miranville & Zelik [12], we set φ α := (1 − α)ϕ + αφ, where α ∈ (0, 1]. Then, assumption (2.2) 4 implies that the function |f (ϕ)| is convex and (3.12) which yields that f (φ α ) ∈ L 1 (Ω). Consequently, φ α is an admissible test function for (3.11). Inserting φ = φ α in the variational inequality (3.11), simplifying by α and using the fact that (ϕ, ψ) is absolutely continuous on [s, t] with values in (3.13) Passing to the limit in (3.13) as α → 0 and using the Lebesgue dominated convergence theorem for the nonlinear term, we obtain: (3.14) We can now state the following theorem which gives the uniqueness of such variational solutions.  1). Furthermore, for every two variational solutions (ϕ 1 , ψ 1 ) and (ϕ 2 , ψ 2 ), we have the following estimate:

15)
where the positive constants c and K are independent of t.
Proof. We use (3.11), with ϕ = ϕ 1 and φ = ϕ 2 , and we obtain: and (3.14) with ϕ = ϕ 2 and φ = ϕ 1 , we find: (3.17) Summing the two resulting inequalities (3.16) and (3.17) and using the fact that (ϕ i , ψ i ) are absolutely continuous on [s, t], i = 1, 2, with values in H −1 (Ω) × L 2 (Γ), we obtain: (3.18) Using the fact that g is bounded globally and applying the Gronwall inequality to (3.18), we have: where the positive constants c and K are independent of t > s > 0 and (ϕ i , ψ i ), i = 1, 2. Passing to the limit as s → 0 and thanks to the continuity of (ϕ i , ψ i ), i = 1, 2 from Definition 3.1, condition 4, we get the desired estimate, which in particular gives the uniqueness. Now, we need to prove that the above definition of a solution is independent of the choice of L. To do so, we assume that (ϕ 1 , ψ 1 ) is a variational solution for L = L 1 and (ϕ 2 , ψ 2 ) is a variational solution for L = L 2 . Using the following relation: and arguing as in the proof of (3.15), we find: (3.21) After simplification, (3.21) gives: which coincides with (3.18) and also leads to (3.15). Theorem 3.1 is thus proven.

23)
for some positive constants α and C which are independent of t and ϕ, where D τ denotes the tangential part of the gradient ∇.
Proof. Repeating the derivation of the variational inequality (3.11), we obtain that (ϕ N , ψ N ) satisfies: for every admissible test function φ and every t > s > 0. Our aim is to pass to the limit N → +∞. We start with the case when the initial datum ϕ 0 is smooth and satisfies the additional conditions: Then, by (2.48), we have: where the positive constant C depends on Ω, Γ, ϕ 0 , ψ 0 and T but is independent of N and t. From this point on, all convergence relations will be intended to hold up to the extraction of suitable subsequences, generally not relabeled. Thus, we observe that weak and weak star compactness results applied to the sequence ϕ N entail that there exists a function ϕ such that as N → ∞, the following properties hold: These convergence results allow us to pass to the limit N → +∞ in (3.24) and prove that the limit function satisfies (3.11) for any admissible test function φ. The only nontrivial term containing the nonlinearity f N can be treated by using the inequality |f N (φ)| ≤ |f (φ)|, the fact that f (φ) ∈ L 1 ([0, T ] × Ω) and the Lebesgue dominated convergence theorem. The crucial point −1 < ϕ(t, x) < 1, for almost all (t, x) ∈ R × Ω, can be proven as in Miranville & Zelik [12]. Indeed, taking into account the definition of f N and the fact that the L 1 −norm of f N (ϕ N ) is uniformly bounded, we can conclude:  N (ϕ N ) → f (ϕ). Therefore, Fatou's lemma gives: ψ) is a variational solution to problem (3.4). In particular, the L 1 −estimate on f (ϕ) follows from (3.39). Since the separation from singularities is not ensured on the boundary, we are not allowed to pass to the limit in ∥F N (ϕ N (t))∥ L 1 (Γ) . Finally, we remove assumption (3.25). In that case, we approximate the initial datum (ϕ 0 , ψ 0 ) ∈ Φ by a sequence (ϕ k 0 , ψ k 0 ) of smooth functions satisfying (3.25) such that: The existence of such a sequence of solutions was proved above. Then, by estimate (3.15) and assumption (3.40), we can see that (ϕ k , ψ k ) is a Cauchy sequence in C([0, T ], H −1 (Ω) × L 2 (Γ)) and therefore, the limit function exists and Then, the proof of the theorem is finished as above.
Concerning the second equation from (3.4), we use Theorem 2.1 and we see that: where the constant c is independent of N . We have: , L 2 (Γ)). Passing to the limit as N → +∞, we have the weak-star convergence in

Additional regularity results and separation from the singularities
In this section, we formulate several sufficient conditions which ensure that every variational solution satisfies equation (3.4) in the usual sense. We have the following result which gives an additional regularity on ϕ close to the points where |ϕ(t, x)| < 1. (3.4). For any δ, T > 0, we set:

Proposition 4.1. Let the assumptions of Theorem 3.1 hold and let (ϕ, ψ) be a variational solution to
Then, ϕ ∈ H 2 (Ω δ (T )) and the following estimate holds: where the positive constant Q δ,T depends on T and δ but is independent of the concrete choice of the solution ϕ.

Corollary 4.2.
Let the assumptions of Theorem 3.1 hold. We assume that: for some ε > 0. Then, for every variational solution ϕ of problem (3.4), estimate (4.12) holds and where the constant C ε,T is independent of the concrete choice of the variational solution ϕ.
Proof. We consider the nonlinear elliptic-parabolic system: Arguing as in Miranville & Zelik [12], we have: where the constant C ε depends on g and ε but is independent of N . Arguing as in Corollary 4.1, we deduce estimate (4.12). To derive (4.21), we multiply (4.22) by f N (ϕ N ) and we use (4.23). We obtain: (4.24) The L 2 -norm of h 1 (t) is controlled thanks to (2.32) and we find: Integrating (4.24) with respect to t and using the fact that f ′ N ≥ 0, we find: (4.26) We deduce from (4.26): Arguing as in Corollary 4.1, we finish the proof.

Attractors and exponential attractors
In this section, we study the asymptotic behavior of the system. We denote by Φ w := H −1 (Ω) × L 2 (Γ). The space Φ w is endowed with the natural norm: We have the following result:

2)
for all (ϕ 1 0 , ψ 1 0 ), (ϕ 2 0 , ψ 2 0 ) ∈ Φ w . This corollary is a direct consequence of Proposition 2.1. The following proposition gives the existence of the global attractor A for this semigroup. We recall that, by definition, a set A ⊂ H −1 (Ω) is the global attractor for the semigroup S(t) if the following properties are satisfied:

Proof.
The semigroup S(t) is dissipative. Indeed, thanks to estimate (2.27) there exists R 0 > 0 such that the ball B H1 (R 0 ) centered on zero with radius R 0 in H 1 (Ω)×H 1 (Γ) is absorbing in Φ w and compact in the topology of Φ w . In particular, there exists a time t 0 ≥ 1 such that S(t)B H1 (R 0 ) ⊂ B H1 (R 0 ), for any t ≥ t 0 . As a consequence, the set: is absorbing and positively invariant. Thus the existence of the global attractor A follows from a proper abstract attractor's existence theorem (see Temam [14]).
In the following theorem, we prove the existence of an exponential attractor which by definition contains the global attractor and has finite fractal dimension. To do this, we first recall the definition of the exponential attractor where A is the global attractor for the semigroup {S(t)} t≥0 : 3. There exist positive constants c 0 and c 1 such that for every u 0 ∈ X, we have: