ON PHASE–FIELD EQUATIONS OF PENROSE––FIFE TYPE WITH THE NON–CONSERVED ORDER PARAMETER UNDER FLUX BOUNDARY CONDITION.

. We study the initial-boundary value problem for the non-conserved phase- ﬁeld model proposed by Penrose and Fife in 1990 [1] under the ﬂux boundary condition for the temperature ﬁeld in higher space dimensions, which is obliged to overcome ad- ditional diﬃculties in the mathematical treatment. In all the existing works concerning this problem, only one due to Horn et al. [2] was discussed under the correct form of the ﬂux boundary condition. Here we prove that the same correctly formulated problem as theirs is well-posed globally-in-time in Sobolev–Slobodetski˘ı spaces. Moreover, it is shown that the temperature keeps positive through the time evolution.


Introduction
In this paper we are concerned with the non-conserved phase-field equation proposed by Penrose and Fife [1,3] which is a continuum model for the description of dynamics of order-disorder phase transition taking into account of both the relaxation and the balance laws based on the second law of thermodynamics: where ϕ is an order parameter, θ is an absolute temperature, M 1 , M 2 and K are positive constants, e = e(ϕ, θ) is the internal energy defined by with the free energy density f = f (ϕ, θ), and g = g(x, t) is a heat supply. The free energy density f (ϕ, θ) assumed in [1,3] is of the form c 2022 Tani A.

A. Tani
with given constants C V > 0, a ≥ 0, δ ≥ 0, b, c and a concave function s 0 (ϕ). The most commonly used form of s 0 is the double equal-well potential, where n = n(x) is an outward unit normal vector to at x ∈ , β is a heat conductivity on the boundary and θ e (> 0) is an external temperature. The latter condition for θ is the so-called flux boundary condition.
A phase transition in two-phase system has been theorized as a continuum in which two phases may coexist, so that the transition between them is considered to occur smoothly within an appropriate layer or diffuse interface. The use of diffuse interface models in describing the phase transition is traced back to van der Waals [4,5], Landau [6] for the second order phase transitions by introducing the notion of order parameter, and Devonshire [7] for the first order transitions. Then, the theory has been extended as continuum models for the description of dynamics of order-disorder phase transition by Cahn [8][9][10] (Models A and B in [11]; see also [12,13]), Caginalp [14], Penrose and Fife [1,3] and so on.
In the sequel, without loss of generality, we assume that C V = M 1 = 1, M 2 = M and β is a positive constant. And it is more convenient to use u = 1/θ instead of θ, so that our problem is formulated as follows: Concerning the mathematical results related to problem (1.6) with β = 0 the existence of a unique strong solution was proved by Zheng [15] and Sprekels and Zheng [16] (see also [17]). In the framework of weak solutions of (1.6) and its generalization there are many papers (see, [18][19][20][21][22] and the references therein). However, for problem (1.6) with β > 0, as was pointed out in [13], all results except [2] were reported under the physically incorrect flux conditions, for example, in [21] M ∂ ∂n and in [22] M ∂θ ∂n = β(θ − θ e ).
The aim of the present paper is to show the unique existence of a strong solution in Sobolev-Slobodetskiȋ spaces in higher space dimensions, which are different from [2]. Moreover, a boundedness of the solution is shown up to an arbitrary finite time.
In what follows, we focus our study on the most important case, N = 3. Let us describe our results.
Assume that the hypotheses in Theorem 1.1 with l = 2, and hold, where | | is a volume of . Then the solution in Theorem 1.1 is extended on [0, T ]: (ϕ, u) W 4,2 2 (QT ) ≤ C(T ), u(x, t) ≥ C * (T ) for any (x, t) ∈ Q T . Here C(T ) and C * (T ) are positive constants depending non-decreasingly on both the data and T . Remark 1.3. Our proof is applicable to the similar problems with more general f , for example, with positive constants C V , α 1 , θ * and nonnegative constants δ 1 , δ 2 , α 3 , α 4 , α 6 satisfying α 4 + α 6 > 0 proposed by Alt and Pawlow [23]. The corresponding results to Theorems 1.1 and 1.2 still hold for such an f .

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is mainly used. For a smooth manifold ∂ = , the spaces W r 2 ( ) of functions defined on are introduced in a standard manner by means of the local coordinates and the partition of unity, and W r,s 2 ( T ), T = × (0, T ), can be defined in the same way as above. The spaces of vector fields whose components belong to, for example, W r,s 2 (Q T ) are denoted by the same notation as the scalar case, W r,s 2 (Q T ), and their norms are supposed to be equal to the sum of norms of all its components. In detail, see [24][25][26].
In what follows, we also denote by c the universal positive constants which may vary in different places; by C i and C i (T ) (i = 1, 2, 3, . . . ) the positive constants depending non-decreasingly on the data but not on T , and may depend non-decreasingly on both the data and T , respectively; besides some constants depend on the indicated quantities.
We give proofs of Theorems 1.1 and 1.2 in Sections 2 and 3, respectively.
2. Local-in-time existence: Proof of Theorem 1.1 ((Step 1)) For simplicity let us assume 1 > l > 1/2, and consider a linear problem are given functions satisfying d(x, t) ≥ d = const > 0 and the compatibility condition.
The theory of linear partial differential equations of parabolic type [25,27] implies that problem (2.1) has a unique solution v ∈ W 2+l,1+l/2 2 Let us denote the extension of (ϕ 0 , u 0 ) by ( ϕ 0 , u 0 ) ∈ W 2+l,1+l/2 2 When we choose in (2.1) In order to estimate f 1 , f 2 , h 1 , h 2 , we rely on the following lemma: By using Lemma 2.1 and interpolation inequalities it is easy to obtain Here l ′ is any number, l ′ ∈ (1/2, l); C 2 and C 3 are positive constants depending on each indicated argument monotonically non-decreasingly; ε i is any positive number; C εi is a positive constant which increases monotonically as ε i tends to 0 (i = 1, 2, 3).
Trace theorem leads to where C 5 , ε 4 and C ε4 have the same properties as C 3 , ε 1 and C ε1 , respectively. Let A 1 and A 2 be any positive constants satisfying respectively. Now we define the successive approximate solution {(ϕ (m) , u (m) )} ∞ m=0 as follows: First we choose ε 1 -ε 4 so small that 4 hold, and then choose T 0 ∈ (0, T ] to satisfy Thus, from (2.7) and (2.8) we conclude by induction for some T * ∈ (0, T 0 ]. For that we begin with the estimates for any t ∈ (0, T 0 ]. Here again ε i is any positive number; C εi is a positive constant increasing monotonically as ε i tends to 0 (i = 5, 6, . . . , 9).
As for (1.8) it is easy to see that holds if t (>)0 is smaller than T 2 = (u 0 /(2A 2 )) 2 . Set T * = min{T 1 , T 2 }, so that the proof of Theorem 1.1 is complete.

Global-in-time existence: Proof of Theorem 1.2
Assume that solution (ϕ, u) to problem (1.6) belongs to W 4,2 2 (Q T ) × W 4,2 2 (Q T ) and has properties (1.7), (1.8) for any T > 0. First we derive a priori estimates of (ϕ, u) on [0, T ]. Some calculations in the following are certainly formal, because the regularity of the solution is not sufficient. However, as usual we can easily derive the rigorous results by using the arguments of mollifiers and passing to the limit.

Then (3.19) becomes
and thus by induction Since it is easily seen that Proof. Differentiate (1.6) 1 with respect to t, and multiply it by ∂ 2 ϕ/∂t 2 . Then by integration by part we have Hence, integrating this over (0, t), we have by virtue of (3.10), (3.12), (3.14) and (3.15) ∂∇ϕ From this it easily follows that by using the equations of (1.6) 1 operated by ∇, ∂/∂t and . For the last estimate in (3.20) we use the inequality which results from integrating (1.6) 2 multiplied by u −2 u over Q t by the help of (3.14)-(3.16).

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Proof. Recall the following equation derived from differentiating (1.6) 2 multiplied by u −2 with respect to t: First, multiply (3.23) by ∂u/∂t and then integrate it over . We get by integration by part and Young's inequality   Combining Lemmas 3.1-3.6 with Theorem 1.1 implies the existence and uniqueness of the solution in Q T for any T > 0 to problem (1.6).
Therefore, the assertion of Theorem 1.2 is proved.