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ON HIGHER-ORDER ANISOTROPIC CAGINALP PHASE-FIELD SYSTEMS WITH POLYNOMIAL NONLINEAR TERMS

Our aim in this paper is to study higher-order (in space) anisotropic Caginalp phase-field systems. In particular, we obtain well-posedness results, as well as the existence of the global attractor and exponential attractor.


Introduction
The Caginalp phase-field system, was proposed in [7] to model phase transition phenomena, such as melting-solidification phenomena. Here, u is the order parameter, T is the relative temperature and f is the derivative of a double-well potential F (a typical choice of potential is F (s) = 1 4 (s 2 − 1) 2 , hence the usual cubic nonlinear term f (s) = s 3 − s). Furthermore, here and below, we set all physical parameters equal to one. This system has been much studied; we refer the reader to, e.g., [2][3][4].
These equations can be derived as follows: One introduces the (total Ginzburg-Landau) free energy where Ω is the domain occupied by the system (we assume here that it is a bounded and regular domain of R 3 , with boundary Γ), and the enthalpy As far as the evolution for the order parameter is concerned, one postulates the relaxation dynamics (with relaxation parameter set equal to one) where D Du denotes a variational derivative with respect to u, which yields (1.1). Then, we have the energy equation where q is the heat flux. Assuming finally the usual Fourier law for heat conduction, we obtain (1.2) In (1.3), the term |∇u| 2 models short-ranged interactions. It is, however, interesting to note that such a term is obtained by truncation of higher-order ones (see [9]); it can also be seen as a first-order approximation of a nonlocal term accounting for long-ranged interactions (see [14]). Now, one essential drawback of the Fourier law is that it predicts that thermal signals propagate at an infinite speed, which violates causality (the so-called paradox of heat conduction). To overcome this drawback, or at least to account for more realistic features, several alternatives to the Fourier law, based, e.g., on the Maxwell-Cattaneo law or recent laws from thermomechanics, have been proposed and studied, in the context of the Caginalp phase-field system, in [20].
In the late 1960's, several authors proposed a heat conduction theory based on two temperatures (see [10,23]). More precisely, one now considers the conductive temperature T and the thermodynamic temperature θ. In particular, for simple materials, these two temperatures are shown to coincide. However, for non-simple materials, they differ and are related as follows: (1.8) The Caginalp system, based on this two temperatures theory and the usual Fourier law, was studied in [4].
Our aim in this paper is to study a variant of the Caginalp phase-field system based on the type III thermomechanics theory with two temperatures recently proposed in [20].
In that case, the free energy reads, in terms of the (relative) thermodynamic temperature θ, and (1.5) yields, in view of (1.8), the following evolution equation for the order parameter: Furthermore, the enthalpy now reads which yields, owing to (1.6), the energy equation Finally, the heat flux is given, in the type III theory with two temperatures, by (see [15,20]) is the conductive thermal displacement. Noting that T = ∂α ∂t , we finally deduce from (1.10) and (1.12)-(1.13) the following variant of the Caginalp phase-field system (see [20]): Caginalp and Esenturk recently proposed in [8] (see also [5,21]) higher-order phase-field models in order to account for anisotropic interfaces (see also [6,17] for other approaches which, however, do not provide an explicit way to compute the anisotropy). More precisely, these authors proposed the following modified (total) free energy and, for β = (0, 0, 0), . This then yields the following evolution equation for the order parameter u: Our aim in this paper was to study the model consisting of the higher-order anisotropic equation (1.18) and the temperature equation (1.16). In particular, we obtain the existence and uniqueness of solutions, as well as the existence of the global attractor and exponential attractors.

Setting of the problem
We consider the following initial and boundary value problem, for k ∈ N: We assume that a β > 0, |β| = k, (2.5) and we introduce the elliptic operator A k defined by where H −k (Ω) is the topological dual of H k 0 (Ω). Furthermore, ((.,.)) denotes the usual L 2 -scalar product, with associated norm . ; more generally, we denote by . X the norm on the Banach space X. We can note that is bilinear, symmetric, continuous and coercive, so that is indeed well defined. It then follows from elliptic regularity results for linear elliptic operators of order 2k (see [1]) that A k is a strictly positive, self-adjoint and unbounded linear operator with compact inverse, with domain We further note that D(A We finally note that (see, e.g., [24] Having this, we rewrite (2.1) as where B 1 = 0 and, for k 2, As far as the nonlinear term f is concerned, we assume that In particular, the usual cubic nonlinear term f (s) = s 3 −s satisfies these assumptions.
Throughout the paper, the same letters c, c and c denote (generally positive) constants which may vary from line to line. Similary, the same letter Q denotes (positive) monotone increasing (with respect to each argument) and continuous functions which may vary from line to line.

A Priori Estimates
We multiply (2.7) by ∂u ∂t and have, integrating over Ω and by parts We then multiply (2.2) by ∂α ∂t − ∆ ∂α ∂t and obtain d dt Summing (3.1) and (3.2), we find the differential equality where B is not necessarily nonnegative). We can note that, owing to the interpolation inequality This yields, employing (2.11) nothing that, owing to Young's inequality, We then multiply (2.7) by u and have, owing to (2.10) and the interpolation inequality (3.5), hence, proceeding as above and employing, in particular, (2.11), Summing (3.3) and δ 1 times (3.9), where δ 1 > 0 is small enough, we obtain a differential inequality of the form Multiplying (2.2) by −∆α, we then obtain (3.12) Summing (3.10) and δ 2 times (3.12), where δ 2 > 0 is small enough, we obtain a differential inequality of the form (3.13) where 14) It particular, if follows from (3.13) − (3.14) and Gronwall's lemma that r > 0, given. Our aim is now to obtain higher-order estimates. To do so, we will distinguish between the cases k = 1 and k 2.
(3.37) Second case: k 2 We multiply (2.7) by A k u and obtain, owing to the the interpolation inequality (3.5) It follows from the continuity of f and F , the continuous embedding H k (Ω) ⊂ C(Ω) (recall that k 2) and (3.15) that Summing (3.13) and (3.40), we find a differential inequality of the form where We then rewrite (2.7) as an elliptic equation, for t > 0 fixed, Multiplying (3.43) by A k u, we have, owing to the interpolation inequality (3.5), hence, owing to (3.39), Next, we differentiate (2.7) with respect to time and obtain ∂ ∂t We finally deduce from (3.45), (3.51) and (3.52) that 53) and (3.54)

The dissipative semigroup
We first have the following theorem.
Then, (u, α) satisfies We multiply (4.2) by u, we obtain, owing to (2.9) and the interpolation inequality Summing then (4.6) and δ 4 times (4.7), where δ 4 > 0 is small enough, we have a differential inequality of the form (note that k 1) where It follows from (4.8), (4.9) and Gronwall's lemma that hence the uniqueness, as well as the continuity (with respect to the L 2 (Ω)×H 2 (Ω) 2norm) with respect to the initial data. We finally turn to the proof of (iii).

Remark 4.2.
We can also prove the continuous depence with respect to the initial data in the H 2k (Ω) × H 2 (Ω) 2 -norm, without any growth restriction on f when k = 1, and it then follows from (3.37), (3.51) and (3.55) that S(t) is defined, continuous and dissipative in H 2k 0 (Ω) × (H 2 (Ω) ∩ H 1 0 (Ω)) 2 . Actually, it follows from (3.36) and (3.53) that S(t) possesses a bounded absorbing set B 1 such that B 1 is bounded in Φ and compact in H 2k (Ω) × H 2 (Ω) 2 . It thus follows from classical results (see, e.g., [19,22,24]) that we have the   (i) We recall that the global attroctor 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 times goes to infinity; it thus appears as a suitable object in view of the study of the asymptotic behavior of the system. We refer the reader to, e.g., [19,22,24] for more details and discussion on this.
(ii) We can also prove, based on standard arguments (see, e.g., [19,22,24]) that A has finite dimension, in the sense of covering dimensions such as the Hausdorff and the fractal dimensions. The finite-dimensionality means, very roughly speaking, that, even thought the initial phase space has infinite dimension, the reduced dynamics can be described by a finite number of parameters (we refer the interested reader to, e.g., [19,22,24] for discussion on this subject).