Behaviour of Mushy Regions under the Action of a Volumetric Heat Source. Daniele Andreucci

Transcription

1 ehaviour of Mushy Regions under the Action of a Volumetric Heat Source Daniele Andreucci* This is a preprint of an article published in Mathematical Methods in the Applied Sciences, 16 (1993), pp Copyright c (1993).G. Teubner Stuttgart John Wiley & Sons, Ltd. *Università di Firenze, Dipartimento di Matematica U. Dini, v.le Morgagni 67/A, 5134 Firenze, Italy. The author is a member of G.N.F.M of Italian C.N.R Postal address: Daniele Andreucci Università di Firenze Dipartimento di Matematica U. Dini v.le Morgagni 67/A 5134 Firenze ITALY Telephone: FISMAT AT IFIIDG Telefax:

2 ehaviour of Mushy Regions under the Action of a Volumetric Heat Source MOS Classification numbers: 35R35, 35K55. Summary. In this paper we consider solutions to Stefan problems in spatial dimension N 1. We find necessary conditions on the heat source for the appearance of a mushy region (i.e., a region where temperature coincides identically with the temperature of change of phase) inside a purely liquid (or solid) phase. For sources depending on energy, such conditions are connected only with the local behaviour of the source near the energy level corresponding to the beginning of the change of phase. oth weak and smooth solutions are considered; in the latter case the behaviour of the solution at the free boundary is investigated in detail. 1. Introduction. In this paper we investigate the behaviour of solutions of problems of change of phase (Stefan problems) in spatial dimension N 1, when a volumetric heat source is present. It is well known that in this case mushy regions may appear in the solution; these are defined as regions of positive measure, where the temperature is identically equal to the critical temperature of change of phase, assumed to be constant. Here we consider both smooth and weak solutions to fairly general Stefan-like problems, giving special attention to the influence of the source on the behaviour of mushy regions. The model problem we have in mind has been investigated in [1], [13]. These authors study the following Stefan problem posed for the one-dimensional heat equation: a domain is occupied at the initial time by a solid phase, and it is heated, for subsequent times, by a positive, constant source. At a certain positive time, the temperature attains the critical value inside the domain. The source being bounded, the change of phase is completed only after a finite time interval has elapsed. Thus, after its appearance, the mushy region keeps expanding into the solid phase, until, eventually, it is invaded by a new liquid phase. Here, as far as regular solutions, defined as in Section 2, are concerned, we relate the behaviour of the solution at the free boundary separating a mushy region from a pure phase, to the behaviour of the source on the free boundary. We also show that our results are sharp. More specifically we show that, under some Lipschitz continuity conditions, a mushy region can expand into a pure phase only if the heat source is non zero, and of suitable sign, on the interface. Moreover, in this case, both the energy and the heat flux are continuous across the interface, contrarily to the case where a pure phase expands into a mushy region. 2

3 Results of this kind were known only in one spatial dimension and under more restrictive assumptions (see [3]), while our main result, Theorem 2.1, is proved under optimal assumptions on the source (see remarks 2.1 and 2.2). Moreover the energy balance equation is taken to be of a quite general form. We also consider weak solutions (see Section 3). In this case, we do not need any regularity of the free boundary. Our results are comparable to the ones given for regular solutions, although less complete, because of the lesser smoothness of the solution, and, in particular, of the free boundary. We also show that, if the source does not satisfy the condition quoted above, the volume of the mushy region does not increase in time. In the special case where the source is zero everywhere, and the space part of the energy equation reduces to the laplacean, results of this type were given in [5], [11], by completely different techniques. On the subject of mushy regions we also quote [1], [6], [9]. Examples of mushy regions disappearing under the effect of a volumetric heat source are known (see e.g., [13]). We give here an example where the new feature is the immediate reappearance of the mushy region after its extinction (see Section 6). In Section 2 we introduce the problem and state the theorems concerning regular solutions. Also, counterexamples showing the sharpness of the results are gathered here. In Section 3, we consider weak solutions to a singular parabolic equation in divergence form. In Sections 4 5 we give the proofs of our results. In Section 6 we construct the example mentioned above. 2. Statement of the problem. Regular solutions. 2.i The problem of change of phase. Regular solutions. A formal statement of the Stefan problem in a domain Ω T bounded open set, T > constant) is Ω (, T ) (Ω R N (2.1) E t (a ij(x, t, u)u xi ) xj + b i (x, t, u)u xi = Q(x, t, E), in Ω T, where u is the temperature. The energy (2.2) E(x, t) = u(x,t) c(x, z) dz + λh(u(x, t)), (x, t) Ω T, (H is the Heaviside graph), has a jump at the temperature u =, chosen as the critical temperature at which the change of phase takes place. The height of the jump, λ > constant, corresponds to the latent heat, while the function c is the heat capacity (the case 3

4 of c depending explicitly on t involves only minor formal complications; it is not treated here for the sake of brevity). In this section we make the standard assumptions (2.3) c > in Ω R; a ij (x, t, s)ξ i ξ j α ξ 2, ξ R N, (x, t, s) Ω T R, with α > given constant; a ij = a ji i, j = 1,..., N. The fact that the temperature of change of phase is constant, is essential in our approach: see Remark 2.3 below. We also remark that, in general, the source Q should be specified as a function of E, rather than of u, in view of the possible existence of regions where u, but E constant. Since our arguments are local we need not prescribe either initial or boundary data. The exact (weak) formulation of free boundary problems of this kind is standard (see Section 3 below). Here we are interested in smooth solutions; more exactly we assume (2.4) u C(Ω T ), and define L = {(x, t) Ω T u(x, t) > }, S = {(x, t) Ω T u(x, t) < }, M = {(x, t) Ω T u(x, t) = }. L and S correspond to the liquid and solid phases respectively, while M (whose measure we assume to be positive) is called a mushy region. We will also assume that, locally, the interface, or free boundary, F separating two of the regions L, S, M has the representation (2.5) F = {(x, t) ψ(x, t) = }, with ψ a function of class C 1, Dψ (we denote D (,..., x 1 x N ) ); we may assume ψ t at points where F is not vertical, that is where F is not tangent to the t axis. We make the further assumptions (2.6) E M C(M); E M t C( M); (2.7) u S x i C(S), u S C 2,1 (S); u L x i C(L), u L C 2,1 (L) i = 1,..., N (the superscript denotes the restriction to the indicated phase). A solution (u, E) fulfilling (2.4) (2.7) will be called a regular solution. Although, to the best of our knowledge, the existence of such solutions in the multidimensional case N > 1 is not known, they are considered here in order to determine the exact behaviour of the temperature and of the energy on the interfaces, avoiding technical complications. We have also chosen to give, for the results of this section, a simple independent proof, though they could be (partially) derived from the results given in next section for weak solutions. 4

5 Since we are not interested here in finding conditions for the existence of regular solutions, but rather in an a priori analysis of their behaviour, for the sake of simplicity we assume (2.8) c S, a S ij, bs i C 1 (Ω T (, ]); c L, a L ij, bl i C 1 (Ω T [, + )); i, j = 1,..., N. We also make the assumption (2.9) Q C(Ω T R); the continuity of Q will be important in our proof, see Remark 2.2. It can be easily seen that, under the assumptions above, a regular solution satisfies in a classical sense (2.1) (2.11) cu t (a ij u xi ) xj + b i u xi = Q(x, t, E), in S L, E t = Q(x, t, E), in M. At the interface F separating two phases A, (A, chosen from L, S, M) we have (2.12) [E] A ψ t = [ a ij u xi ψ xj ] A ; (2.12) reduces to the classical Stefan condition if A = S, = L. 2.ii ehaviour of the solution at the free boundaries and growth of mushy regions. In order to prove our main result, we assume the following: (2.13) (x o, t o ) Ω T ρ, β > such that if Q(x o, t o, ) =, then Q(x, t, s) βs in ρ (x o, t o ) ( ρ, ) (here ρ (x o, t o ) { (x, t) (x, t) (x o, t o ) < ρ }). Note that (2.13) does not impose any restriction if Q(x o, t o, ). An example of a source satisfying (2.13) is Q(x, t, E) = f(x, t)g(e), with f bounded in Ω T and g Lipschitz continuous at, g() =. With the notation introduced in (2.5) we define the normal velocity of F at (x, t) by v n (x, t) ψ t (x, t) Dψ(x, t) 1. If, in particular, F separates M and S, and we have locally (2.14) M= {ψ > }, S = {ψ < }, the interface advances in S (that is M grows) if and only if v n <. The following theorem, under the assumptions above, relates precisely the behaviour of u and Q on F to the motion of the interface. 5

6 Theorem 2.1. Let (u, E) be a regular solution to the Stefan problem (2.1) (2.2) in Ω T, and assume that (2.3), (2.8) (2.9), (2.13) are satisfied. If the convention (2.14) is in force, then (x, t) F, a) v n (x, t) > Du S (x, t), E M (x, t) > ; b) v n (x, t) Du S (x, t) =, Q(x, t, ) > ; c) v n (x, t) < E M (x, t) =, where D ( x 1,..., x N ). Remark 2.1. Assumption (2.13) is necessary for the full validity of Theorem 2.1 (but see also Remark 2.2): if (2.13) does not hold, a mushy region can grow even if the source is zero on the free boundary. Indeed, the function σ (, 1) fixed, solves U(x, t) = (x t) 2 1 σ +, x R, t >, U t U xx = 2(1 σ) 1 [x t + (1 + σ)(1 σ) 1 ] U σ, in S = {U < } = {x > t > }, and E U in M = {t >, t x}. It is easily seen that U is a solution to a Stefan problem with an expanding mushy region and a (non Lipschitz continuous) heat source vanishing at the interface. Remark 2.2. It follows from the proof of Theorem 2.1 that the continuity of Q in the variable E across E = is actually needed only in part a), and only if Q(x, t, ) >. If this assumption is dropped, examples contradicting the first part (and therefore also the second part) of implication a) are immediately found. For instance, (V, E), with V = (x + t) 2 +, E(V ) = V + λh(v ), λ >, solves E t V xx = 2(x + t) sgn E, x R, t >, where sgn (s) 1 if s <, sgn (s) if s. The solid region S = {x + t > } expands into the mushy region M = {x + t } (where the energy is assumed to be identically zero), though V = V x = at the interface {x + t = }. ut notice that the source is discontinuous across the interface: indeed V t V xx 1 in {1/2 > x + t > }. It also follows from the proof, that the first part of b), and c), hold even if assumption (2.13) is removed. The case of an interface L/M is similar: we have to assume (2.13 ) (x o, t o ) Ω T ρ, β > such that if Q(x o, t o, λ) =, then Q(x, t, s) β(s λ) in ρ (x o, t o ) (λ, λ + ρ). Theorem 2.2. Under the assumptions of theorem 2.1, with (2.13) replaced by (2.13 ), and (2.14) replaced by (2.14 ) M= {ψ > }, L = {ψ < }, 6

7 we have (x, t) F a) v n (x, t) > Du L (x, t), E M (x, t) < λ; b) v n (x, t) Du L (x, t) =, Q(x, t, λ) < ; c) v n (x, t) < E M (x, t) = λ. Remark 2.3. In [3] it is shown via a counterexample that if the temperature of change of phase is not constant, mushy regions can develop even in the absence of sources. Theorem 2.1 will be proved in Section 4; Theorem 2.2 can be proved analogously. 3. Weak solutions. (3.1) In this section we consider the equation E t div a(x, t, u, Du) = Q(x, t, E), in Ω T ; E has been defined in (2.2); in this section we choose c 1, λ 1, for the sake of simplicity; see also Remark 5.1. We stipulate the following structure assumptions (3.2) a(x, t, s, p) C p, Q(x, t, s) C, a(x, t, s, p) p C 1 p 2, (x, t, s, p) Ω T R N+1. We will also use, in some of our results, the assumption (3.3) Q(x, t, s) C 1 s, (x, t, s) Ω T (1 C 1, 1 + C 1 ) ; here C > 1 is a given constant; (3.3) is a stronger version of assumption (2.13 ). We make requirement (3.3) on the behaviour of Q near E = 1 because we are going to deal in this section with the case of a liquid phase coexisting with a mushy region (i.e., we assume u ). Results and proofs can be straightforwardly rephrased to cover the case u. Definition 3.1. A pair (u, E), E = u + w, with u L (Ω T ) W 1 2 (Ω T ), w L (Ω T ), w H(u) a.e. in Ω T, is a local solution to (3.1) in Ω T if (x, t) a(x, t, u(x, t), Du(x, t)), (x, t) Q(x, t, E(x, t)) are measurable and, for a.e. < t < t 1 < T (3.4) Ω E(x, t)η(x, t) dx t=t 1 t 1 t=t t Ω for any η W 1 2 (Ω T ) L (Ω T ), η near Ω (, T ). { Eηt a(x, t, u, Du) Dη + Q(x, t, E)η } dx dt =, 7

8 A pair (u, E) satisfying the requirements of Definition 3.1, excepting the regularity assumption u W2 1(Ω T ), replaced by u V 1, 2 (Ω T ) L 2(, T ; W2 1(Ω T ) ) C (, T ; L 2 (Ω) ), is called a solution of class V 1, 2 (Ω T ). Existence of weak solutions to problems of Stefan type has been the subject of many investigations; see for example [5], [8], [12]. In [2] it is proved that any solution in the sense of Definition 3.1 is continuous in Ω T. This fact will be used in the sequel. Theorem 3.1 below is the main result of this section. Theorem 3.1. Let (u, E) be a solution to (3.1) in Ω T, and let u in Ω T. Let (3.2) hold. Then (3.5) w(x, t) w(x, τ) t τ Q(x, s, w(x, s))χ {u=} (x, s) ds, a.e. in Ω T, τ < t. Moreover, let (3.3) hold. Then, for a.e. < τ < t < T, (3.6) meas ( {x Ω u(x, t) = } ) meas ( {x Ω u(x, τ) = } ). Since our methods are entirely local, if we know only u in G (, T ), where G is a subdomain of Ω, Theorem 3.1 still holds in G (, T ). It has been shown in Remark 2.1 that (3.6) may not hold if Q is not Lipschitz continuous. Inequality (3.5) is a weak analogue of part c) of Theorem 2.2. Indeed, under the regularity assumptions of Section 2, (3.5) is easily seen to imply the continuity of E across the interface. Remark 3.1. Let u be as in Theorem 3.1, and let u(x, τ) >, τ > given, for all x Ω. If (3.2) and the weaker form of (3.3) Q(x, t, s) C(1 s), s (1 C 1, 1) hold, then it follows from (3.5) that w 1 in Ω (τ, T ), even if a mushy region appears. This is indeed the case in the counterexample given in Remark 2.1. The existence of the time derivative u t as an integrable function is connected to some regularity of the constitutive functions in equation (3.1) (see [8], [12]). In [2] it is shown that solutions of class V 1, 2 (Ω T ) are also continuous, provided they can be approximated in a weak sense by solutions to suitably regularized problems. y approximation we may prove a result analogous to Theorem 3.1 for solutions in V 1, 2 (Ω T ). To this purpose we introduce a sequence of problems E n t div a n (x, t, u n, Du n ) = Q n (x, t, E n ), in Ω T, 8

9 n = 1, 2,... (where a n, Q n fulfil (3.2)), whose solutions (u n, E n ) (in the sense of Definition 3.1) satisfy as n (3.7) u n u in L 2 (Ω T ) ; and w n w, Q n (x, t, E n ) Q(x, t, E), a n (x, t, u n, Du n ) a(x, t, u, Du), weakly in L 2 (Ω T ). Theorem 3.2. Let (u, E) be a solution to (3.1) of class V 1, 2 (Ω T ). Let (3.2) hold. Assume also that (u, E) is the limit in the sense of (3.7) of a sequence {(u n, E n )} as above. If u n in Ω T, n = 1, 2,..., then (3.5) holds. If moreover, Q satisfies (3.3) and u is continuous in Ω T, (3.6) holds. 4. Proof of Theorem 2.1. Proof of a): We proceed by contradiction, proving that for any (x o, t o ) F, (4.1) Du S (x o, t o ) = implies (4.2) ψ t (x o, t o ). If Q(x o, t o, ) we have by assumption (2.13) (4.3) cu t (a ij u xi ) xj + b i u xi + βku = Q(x, t, E) + βku, in a domain S δ (x o, t o ), for δ > small enough; in (4.3) β is the same constant of (2.13), and k is a local bound for c. On applying the boundary point principle to the operator in (4.3), we find (4.4) Du S (x o, t o ) Dψ(x o, t o ) >, contradicting (4.1). Then we conclude (4.5) Q(x o, t o, ) >. Now notice that (2.12) and (4.1) imply (4.6) E M (x o, t o )ψ t (x o, t o ) =. If ψ t (x o, t o ) =, there is nothing to prove; hence we may assume (4.7) E M (x o, t o ) =, ψ t (x o, t o ). ut, in M, E obeys the ordinary differential equation (2.11). Then, if ψ t (x o, t o ) <, for some ɛ > the segment {(x o, τ) τ (t o ɛ, t o )}, 9

10 is contained in M; thus (4.5) and (4.7) together with (2.11) would imply E < somewhere in M (because Q is continuous in E), contradicting the definition of mushy region. Therefore ψ t (x o, t o ) >. We have proved the first part of a). The second part follows easily from (2.12) and the first part. Proof of b): Clearly a S ij us x i ψ xj on F. Then we infer from (2.12) that (4.2) implies a S ij us x i ψ xj = at (x o, t o ). It follows easily that (4.1) holds; finally, it has been shown above that (4.1) implies (4.5). Proof of c): This is an obvious consequence of part b) and (2.12). 5. Proofs of Theorem 3.1 and Theorem 3.2. Proof of Theorem 3.1: Let be any ball contained in Ω. In the following we denote (t) = {t}. For the sake of notational simplicity, we may assume without loss of generality that u is defined in an open neighbourhood of [, T ]. We also remark that from Definition 3.1 it follows that t Ω w(x, t)η(x) dx is continuous for any η L2 (Ω) (after being redefined in a subset of (, T ) of measure zero). We use in (3.4) test functions of the form where ζ Co (), ζ, and β, ε (, 1). We get, after standard calculations, (5.1) 1 1 β + β + 2 t (τ) t for any t (, T ). Note that (u + ε) 1 β ζ 2 dx u τ η(x, t) = (u(x, t) + ε) β ζ 2 (x), τ= + w (u + ε) 1+β ζ2 dx dτ β a Dζ(u + ε) β ζ dx dτ = u τ (τ) t t w(u + ε) β ζ 2 dx a τ= Du (u + ε) 1+β ζ2 dx dτ Q(x, t, E)(u + ε) β ζ 2 dx dτ, (5.2) w (u + ε) 1+β = u τ, in (, T ) ; (u + ε) 1+β 1

11 indeed both sides of (5.2) equal zero a.e. in {u = }, and, on the other hand, w = 1 a.e. in {u > }. Therefore (5.3) β t = u τ w (u + ε) 1+β ζ2 dx dτ + (τ) (τ) (w 1)(u + ε) β ζ 2 dx w(u + ε) β ζ 2 dx τ= = ε β wζ 2 dx where we have used again w = 1 a.e. in {u > }. Then, employing assumption (3.2) and Cauchy-Schwartz inequality, we estimate the space part τ= (τ) τ=, (5.4) β t a βc 1 β 2 C 1 γ(ζ) Du (u + ε) 1+β ζ2 dx dτ + 2 t t t t Du 2 (u + ε) 1+β ζ2 dx dτ + 2C a Dζ(u + ε) β ζ dx dτ t Du 2 (u + ε) 1+β ζ2 dx dτ + 2 β C2 t Du Dζ (u + ε) β ζ dx dτ Dζ 2 (u + ε) 1 β dx dτ (u + ε) 1 β dx dτ, γ(ζ) = 2C 2 β 1 Dζ 2,. (5.5) From (5.1), (5.3) (5.4) we get for t (, T ) 1 1 β (τ) + γ(ζ) (u + ε) 1 β ζ 2 dx t τ= (u + ε) 1 β dx dτ + ε β t (τ) wζ 2 dx τ= Q(x, t, E) (u + ε) β ζ2 dx dτ. First we prove (3.5). We multiply both sides of (5.5) by ε β, and let ε, keeping β fixed. Note that, as ε, (5.6) ε β (u + ε) β χ {u=}, a.e. in (, T ). 11

12 Thus we get for t (, T ), as a result of the limiting process, (5.7) (τ) wζ 2 dx τ= t Q(x, t, w)χ {u=} ζ 2 dx dτ. From (5.7) and the arbitrariness of ζ and, (3.5) follows. Finally, we turn to the proof of (3.6). Since u is continuous (see [2]), it will be enough to show that for any ball Ω (5.8) u(x, ) >, a.e. x = u(x, t) >, a.e. x, t (, T ). From (3.5) and assumption (3.3), we have for t (, T ) (5.9) w dx w dx t Q(x, t, w)χ {u=} dx (t) () C t (1 w) dx dτ. If u > a.e. in (), then w = 1 a.e. in (), implying, together with (5.9), w = 1 a.e. in (, T ). Therefore, the term in (5.5) containing ε β vanishes, and, by assumption (3.3), β, ε (, 1) Q(x, t, E) (u + ε) β γ u o (u + ε) β, for some constant γ o >. Then we may let ε in (5.5) to find (5.1) u 1 β ζ 2 dx u 1 β ζ 2 dx + (1 β)γ(ζ) t u 1 β dx dτ (t) () γ o (1 β) t u 1 β ζ 2 dx dτ. Then we let β 1 in (5.1), getting (5.11) χ {u>} ζ 2 dx (t) () χ {u>} ζ 2 dx, t (, T ). Since ζ C o (), ζ is arbitrary, implication (5.8) is proved. 12

13 Proof of Theorem 3.2: We remark that the assumption u t L 2 (Ω T ) has been used only to prove (5.5). Thus we only need show that (5.5) holds under the assumptions of Theorem 3.2. For n = 1, 2,..., we have for < ϑ < t < T (5.12) 1 1 β (τ) + γ(ζ) (u n + ε) 1 β ζ 2 dx t ϑ τ=ϑ (u n + ε) 1 β dx dτ + ε β t ϑ (τ) w n ζ 2 dx τ=ϑ Q n (x, t, E n ) (u n + ε) β ζ2 dx dτ. Integrate both sides of (5.12) in t over (t o, t o + h), and in ϑ over (ϑ o, ϑ o + h); here < ϑ o < t o < T are arbitrarily chosen, and t o ϑ o > h >. Then we may take the limit n on both sides of the inequality so obtained, by virtue of (3.7), getting 1 1 β ϑ o +h ϑ o + γ(ζ) t o +h t o ϑ o +h ϑ o (τ) t o +h t t o (u + ε) 1 β ζ 2 dx ϑ τ=ϑ dtdϑ + ε β (u + ε) 1 β dx dτ dtdϑ ϑ o +h ϑ o ϑ o +h ϑ o t o +h t o t o +h t t o ϑ (τ) wζ 2 dx τ=ϑ dtdϑ Q(x, t, E) (u + ε) β ζ2 dx dτ dtdϑ. Finally we divide both sides of this inequality by h 2 and let h, finding for < ϑ o < t o < T the analogue of (5.5). Then the proof can be completed as before. Remark 5.1. It is apparent from the proofs that theorems 3.1 and 3.2 continue to hold for supersolutions of (3.1) (provided they are continuous). These are defined as pairs (u, E) satisfying Definition 3.1, excepting equality (3.4): we require instead the left hand side of (3.4) to be non negative if η. Moreover the case c = c(u), < C 1 < c < C, can be treated as above, simply by using test functions η ( u c(s) ds + ε) β ζ A model problem. Model problems of type (2.1) (2.2), showing the appearance of mushy regions, are well known in the literature ([7], [1]), at least in one spatial dimension. Examples of mushy regions eventually disappearing under the action of a distributed source are also known ([13]). Here we give a new example illustrating the possible behaviour of a (regular) solution immediately after the disappearance of a mushy region. In [13] an example is considered of such a region, enclosed between a solid and a liquid phase, disappearing at a time t >, 13

14 and not reappearing for t > t, where a classical two-phase Stefan problem is to be solved; in that example the source is assumed to be constant. The example we find here shows that, still assuming the source to be constant, a mushy region can reappear immediately after its extinction. Namely, we construct a problem whose solution exhibits a mushy region, enclosed by two solid phases, disappearing at t = t >, and reappearing immediately after. In contrast with the previous example, the free boundaries are tangent at the common point for t = t ; this fact, and the results of [4], allow us to complete the proof. We start considering the (classical) solution of the Stefan problem (6.1) (6.2) (6.3) (6.4) (6.5) (6.6) u t u xx = Q, in D T = { < t < T, < x < s(t)}, u x (, t) = r, < t < T, u(x, ) = r(x b), < x < b, u(s(t), t) =, < t < T, u x (s(t), t) = Cṡ(t), < t < T, s() = b. Here T > is to be chosen, and Q, r, b, C are positive constants satisfying the conditions (6.7) Qb > r, C > Qrb 2 /2(Qb r), which will be used later. We remark that (6.1) (6.6) is a problem of change of phase in the sense of (2.1) (2.2), as long as u < in D T : D T is the solid phase and {x > s(t)} is the mushy region. Condition (6.5) and the relation between C and the latent heat λ will be commented below. A standard application of the maximum principle to u x yields u x >, u < in D T, for any T > such that ṡ(t) > t (, T ). As a consequence of Gauss-Green theorem, we have for any t > t (6.8) r b2 2 + rt C ṡ(τ) dτ + s(t) t u(x, t) dx = Q s(τ) dτ. This equality, together with (6.7), would lead us to a contradiction if we had ṡ(t) > for every t >. Then, setting we get from (6.8) T inf{t > ṡ(t) = }, s o s(t ), (6.9) T < rb 2 /2(Qb r). We define u by symmetry around the axis x = s o, in the domain D T = {(x, t) 2s o s(t) < x < 2s o, < t < T }, 14

15 and extend u to zero in the region M T R \ (D T D T ), where R = (, 2s o) (, T ). The function u obtained solves the Stefan problem (6.1) (6.11) (6.12) E t u xx = Q, in R, E(x, ) = E o (x), x (, 2s o ), u x (, t) = u x (2s o, t) = r, t (, T ), where E(u) = u + λh(u), and E o must be suitably prescribed (in (b, 2s o b)), so that (2.11) holds in M T, and (2.12) reduces to (6.5) at the free boundary. This amounts to defining E o (x) C Qs 1 (x), for x (b, s o ), with s 1 inverse function of s. E o is an acceptable datum if λ > E o >. The first inequality is guaranteed by λ > C, which can be assumed. The second one follows from (6.7), (6.9) and C Qs 1 (x) C QT. In (s o, 2s o b) E o is defined by symmetry. The solution to the problem (6.1) (6.12) can be continued for t > T ; such a continuation will be still denoted by u. Incidentally, we notice that also the solution to (6.1) (6.6) can be continued for t > T, but it is different from u; to see this, just note that the flux at the free boundary is not zero, even when the boundary is receding into the solid region. At (s o, T ) the two components of the free boundary are tangent, and the mushy region disappears. We now show that it immediately reappears for t > T. The function v = u t is the unique bounded solution to (6.13) (6.14) (6.15) (6.16) v t v xx =, in D T, v(x, ) = Q, < x < b, v x (, t) =, < t < T, v(s(t), t) = Cṡ(t) 2, < t < T. It is apparent that a level curve v = originates from (b, ). The maximum principle and the analyticity of v as a function of x, guarantee that only one curve v = exists in D T ; we denote it by Γ. Let G be the domain bounded in D T by Γ and x = s(t). If dist(γ, (s o, T )) >, the boundary point principle and (6.16) would imply v x (s o, T ) >. On the other hand, by definition of T, s(t ), and C s(t ) = u xx (s o, T )ṡ(t ) + u xt (s o, T ) = v x (s o, T ), a contradiction. Thus Γ hits (s o, T ). Since, as we have seen, v = u t is positive in D T \ G, we have proved (6.17) u xx (x, T ) + Q = u t (x, T ) >, x (, s o ) (s o, 2s o ). This is exactly the condition required by [3 thm. 3.1] to ensure the appearance of a mushy region at (s o, T ), expanding for increasing t. For the sake of brevity we do not reproduce the proof here. We only say that, in a right neighbourhood of T, the mushy region is the domain bounded by two continuous curves σ 1, σ 2, with σ 1 decreasing, σ 2 increasing, and σ 1 (T ) = σ 2 (T ) = s o. 15

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