ASYMPTOTIC BEHAVIOR OF A CAHN-HILLIARD/ALLEN-CAHN SYSTEM WITH TEMPERATURE

. The main goal of this paper is to study the asymptotic behavior of a coupled Cahn-Hilliard/Allen-Cahn system with temperature. The work is divided into two parts: In the ﬁrst part, the heat equation is based on the usual Fourier law. In the second one, it is based on the type III heat conduction law. In both parts, we prove the existence of exponential attractors and, therefore, of ﬁnite-dimensional global attractors. following

1. Introduction. J. Cahn and A. Novick-Cohen introduced, in [3], the following system: where u is the concentration of one of the components and it is a conserved quantity, v is an order parameter, h is a (positive) parameter which represents the lattice spacing, α is a parameter that reflects the location of the system within the phase diagram (it may be either positive or negative), and the nonlinear term f is the derivative of a double-well potential F . The system models simultaneous order-disorder and phase separation in binary alloys on a BCC lattice in the neighborhood of the triple point.
We further note that it is a gradient flow in H 1 × L 2 for the free energy

ASYMPTOTIC BEHAVIOR OF A CAHN-HILLIARD/ALLEN-CAHN SYSTEM 2259
Indeed, we can rewrite this equation as where q is the thermal flux vector and, assuming the Fourrier law The function H is the enthalpy defined by and θ is the relative temperature.
Summing (10), (11), and (12), we obtain Based on (13), we have the following result Theorem 2.1. Assume that (u 0 , v 0 , θ 0 ) ∈ H 1 0 (Ω) 3 . Then, (1)-(5) possesses at least one solution (u, v, θ) such that (u, v, θ) ∈ L ∞ (IR + ; H 1 0 (Ω) 2 × L 2 (Ω)) ∩ L 2 loc (IR + ; H 2 (Ω) 3 ), θ ∈ L ∞ (IR + ; L 2 (Ω) ∩ H 1 0 (Ω)) and ( ∂u ∂t , ∂v ∂t , ∂θ ∂t ) ∈ L 2 (IR + ; H −1 × L 2 (Ω) 2 ). Proof. The proof of existence (as well as the above and the subsequent a priori estimates) are based, e.g. on a classical Galerkin scheme. Let A denote the minus Laplace operator associated with Dirichlet boundary conditions. This operator is a bounded, selfadjoint and strictly positive operator with compact inverse from H 1 0 (Ω) onto H −1 (Ω). There is a set of eigenvectors {φ i , i ≥ 1} for this operator, associated with the eigenfunctions 0 < λ 1 ≤ λ 2 ≤ ..., such that it is orthonormal relative to the inner product in L 2 (Ω) and orthogonal relative to the one in H 1 0 (Ω). Setting V m = Span{φ 1 , ..., φ m }, we consider the following approximating problem, written in the fuctional form: together with suitable initial conditions, namely, where P m is the orthogonal projector from L 2 (Ω) onto V m (for the L 2 −metric). This is equivalent to the following problem: ∀ p, q, r ∈ V m , together with the above initial conditions. The proof of existence of a local (in time) solution to the approximating problem is standard (indeed, one has to solve a continuous system of ODEs). Furthermore, we can write the equivalent of the previous and the subsequent estimates (with u, v and θ replaced by u m , v m , and θ m respectively); this is now fully justified and no longer formal. Then we can deduce from (13) that this solution is actually global. And, the passage to the limit is based on classical (Aubin-Lions type) compactness results. Indeed, we have, in particular, u m bounded in L ∞ (0, T ; H 1 0 (Ω)) and dum dt bounded in L 2 (0, T ; H −1 (Ω)), independently of m, which yields that (at least for a subsequence which we do not relabel) u m converges strongly to, say, u in C([0, T ]; H 1−δ (Ω)), ∀ δ > 0. In addition, v m is bounded in L ∞ (0, T ; H 1 0 (Ω)) and dvm dt is bounded in L 2 (0, T ; L 2 (Ω)), independently of m, which also yields the strong convergence of v m to, say, v in C([0, T ]; H 1−δ (Ω)), ∀ δ > 0.
Lemma 2.3. The same solution also verifies: a) Proof. a) We start by recalling the inequality (20) Using the interpolation inequality and then the Young's inequality, we obtain Applying now Gronwall's lemma and using (23), we find b) It follows from (53) after using (23) and (50) Hence, there exists T ∈ (0, 1) such that (55) Repeating the estimates leading to (49), but starting from t = T instead of t = 0, we have Then using (55), we obtain We now repeat the estimates leading to (56), and since our equations are autonomous, we can make a translation in time. We obtain, for t ≥ 1, which yields, owing to (22), Combining the above estimate with (49) from 0 to 1, we obtain Furthermore, we recall the equation (27) d dt Noting that it follows from (23), and (78) that and from (47), (58) and (61) that We deduce from (60)-(62) and the uniform's Gronwall lemma, (see, e.g. [39]), that Collecting (59) and (63), we obtain Proof. The proof of the existence (as well as the above a priori estimates) are based on a classical Galerkin scheme as in the previous section and mainly on the estimates (27) and (64).
We then have the following system We multiply (65) by ∂u ∂t , (66) by ∂v ∂t , and (67) by θ. We then sum the result to obtain Furthermore, and similarly Therefore, Owing to (64), we can see that In the same way, where We deduce from (70)-(75) that Now using Gronwall's lemma, we obtain whence the uniqueness (taking (u 0 , v 0 , θ 0 ) = (0, 0, 0)), as well as the continuous dependence with respect to the initial data.
2.3. Global and exponential attractors. We set E = (H 2 (Ω) ∩ H 1 0 (Ω)) 3 . Note that it follows from Theorem 2.2 that we can define the semigroup is the unique solution to our system.   (iv) M attracts exponentially fast the bounded subsets of E : where the constant c is independent of B and dist H 1 (Ω) 2 ×L 2 (Ω) denotes the Hausdorff semidistance between sets defined by Proof. Here, we assume that the initial conditions are in the bounded absorbing set B 0 . To complete the proof, we need an asymptotic smoothing property on the difference of two solutions, a Hölder estimate with respect to space and time, and a compactness estimate of the solution. These are the key tools to construct exponential attractors (see [7]- [9], [10,25], and [26]).
The Hölder estimate is as follows We now differentiate equations (2) and (9) with respect to time, then we multiply the resulting equations by ∂u ∂t and ∂v ∂t respectively, we use (7) and an interpolation inequality to find (78) We note that it follows from (23), (24), (78) and the fact that the initial conditions are in a bounded absorbing set that t2 t1 ∇ ∂u ∂t where c only depends on B 0 and T ≥ T 0 such that t 1 , t 2 ∈ [T 0 , T ]. Moreover, it follows from (76) and (77) where c only depends on B 0 . Plus, it follows from (21) where c only depends on B 0 . Therefore, we have where c only depends on B 0 , and t 1 , t 2 ∈ [T 0 , T ], where T ∈ IR + . We now want to find a compactness estimate: First, we differentiate (65) and (66) with respect to time, we multiply the resulting equations by (t − T 0 ) ∂u ∂t and (t − T 0 ) ∂v ∂t respectively, where T 0 is the same as before and we obtain Noting that using Hölder inequality and the continuous embeddings H 1 (Ω) ⊂ L 3 (Ω) and H 1 (Ω) ⊂ L 6 (Ω). Similarly, Owing then to a proper interpolation inequality, we obtain Noting also that it follows from (23) for (u, v) = (u 2 , v 2 ) and the constants only depend on B 0 . Furthermore, applying Gronwall's lemma on (85) over (T 0 , t) and owing to (77), (80), (81) and (86), we obtain where the constants only depend on B 0 . Next, we rewrite equations (65) and (66) in the following forms and satisfy, owing to (77) and (87), and where the constants only depend on B 0 . Multiplying now (88) by −∆u and (89) by −∆v, we obtain where the constants only depend on B 0 .
We also multiply (67) by −(t − T 0 )∆θ and find d dt We find combining (85) and (95) then applying Gronwall's lemma (applied over (T 0 , t); note that T 0 ≤ 1) and using (77)-(81), and (86), we find And the result follows from (77) Remark 2. The global attractor A is the smallest (for the inclusion) compact set of the phase space which is invariant by the flow (i.e., S(t)A = A, ∀t ≥ 0) and attracts all bounded sets of initial data as time goes to infinity; that's why, it's important in the study of the asymptotic behavior of the system. Furthermore, the finite dimensionality means, roughly speaking, that, even though the initial phase space is infinite dimensional, the reduced dynamics can be described by a finite number of parameters. We refer the reader to [1,37,39], and [26] for more details and discussions on this topic.
3. Part II: The type III law. The classical Fourier law presented in the previous section has one essential drawback, that is, it predicts that thermal signals propagate with an infinite speed, which violates causality (the so-called 'paradox of heat conduction', see, e.g., [4]). That is why, several modifications of this law have been proposed in the literature to correct this unrealistic feature, leading to a second order in time equation for the temperature.
In particular, in [22], the authors considered in the place of the Fourier law, the Maxwell-Cattaneo law 1 + η ∂ ∂t q = −∇θ, η > 0, which leads to (see also [15,16]). On the other hand, Green and Naghdi proposed in [11]- [14] an alternative treatment for a thermomechanical theory of deformable media which presents an entropy balance rather than the usual entropy inequality. However, if we restrict our attention to the heat conduction, we recall that three different theories, labelled as type I, type II and type III, were proposed. In particular, the Fourier law is found when type I is linearized. The linearized versions of the two other theories are described by the following constitutive equations (knowing that we are going to study only the type III in what follows) and q = −k∇α − k * ∇θ, k, k * > 0, (Type III), (99) where α(t) = t t0 θ(τ )dτ + α 0 , θ = ∂α ∂t is called the thermal displacement variable. These theories were well studied in the recent years and, particularly, a special interest was given to the qualitative study of the solutions (see e.g. [29]- [34] for studies concerned with linear thermoelastic theories). In addition, non-linear acceleration waves have been studied for types II and III non-linear thermoelasticity [35] and fluids without energy dissipation [36].
Adding equations (98) and (99) to the equation we obtain the following equations 3.1. Setting of the new problem. We consider the following initial and boundary value problem (for simplicity, we take k = k * = 1): where Ω is a bounded domain of IR N (N = 1, 2, or 3) with smooth boundary Γ.
We assume that f is of class C 2 and satisfies where We also assume that 3.2. Global and exponential attractors. where e) Proof. a) We rewrite (101) in the equivalent form We multiply (113) by ∂u ∂t , (102) by ∂v ∂t and have, summing the results, ∂u ∂t We then multiply (103) Then, we multiply (103) by α and obtain d dt We sum (108), 1 (116) and 2 (117), where 1 and 2 > 0 are chosen small enough so that and have an inequality of the form where In particular, we deduce from (121) the following estimate Furthermore, for every r > 0, c) We multiply (101) by ∂u ∂t , (102) by −∆ ∂v ∂t , and (103) by −∆ ∂α ∂t , we obtain d dt and Summing (124), (125), and (126) yields In particular, we set Thus, we deduce from (127) a differential inequality of the form y ≤ Q(y).
Let z be the solution of the ordinary differential equation with z(0) = y(0). It follows from the comparison principle that there exists Therefore (u, v, α, ∂α ∂t ) ∈ L ∞ (0, T 0 ; H 2 (Ω) 3 × H 1 (Ω)) a priori. We now differentiate (102) and (113) with respect to time to find, owing to (103) Multiplying then (129) by t ∂u ∂t , (130) by t ∂v ∂t , and (103) by t ∂α ∂t to obtain, summing the three resulting inequalities and using (107) and an interpolation inequality, Moreover, we note that it follows from (127) and (155) that Multiplying then (103) by −∆α, we have We now sum (121) and 3 (149), where 3 > 0 is small enough so that where We also sum (156) where We finally deduce from (159) the inequality ∂α ∂t ∈ L ∞ (IR + ; H 1 0 (Ω)) ∩ L 2 (IR + ; H 2 (Ω)). Proof. The proof of existence is based on the a priori estimates mentioned in the previous lemmas and on, e.g., a standard Galerkin scheme similar to the proof of Theorem 2.1 based mainly on (108) and (112). Therefore, we will only be proving the uniqueness.
On the other hand, where c only depends on B 1 and T ≥ T 0 such that t 1 , t 2 ∈ [T 0 , T ].. Moreover, looking at the equation (162), we can see that Thus, it follows from (111), (112), (184), and (185) that t2 t1 ∂ 2 α ∂t 2 ≤ c, where c only depends on B 1 . Whence, we have It is important to note that w 2 , q 2 , and r 2 verify the same system as u, v, and α so they satisfy the same estimations found in the previous sections. Furthermore, we differentiate (175) and (176)  Consequently, we deduce from standard results the Corollary 2. The semigroup S(t) possesses the finite dimensional global attractor A ⊂ B 1 .