A free-boundary problem for a degenerate parabolic system

JOURNAI. OF DlFPEREN’I’IAl~ EQUATlONS 50, I-1 Y (1983) A Free-Boundary Problem for a Degenerate Parabolic System lJkv?rsily Mathematics Research Center. qj’ Wisconsin, .bfudison. Wisconsin 53706 AND R. E. SHOWALTER Department R1.M Y.100. The lJnioersi/,g qf Mathematics. OJ’Texas, Austin, Tcxns 78712 Rcccived August 17, 1981 I. INTRC)D~JCTION We shall be concerned with the first initial-boundary-vaiue problem for non-negative solutions of a system of nonlinear partial differential equations of the form $w, I)) ---‘@0(.x, I) -t-h($& f) 0(x, t)) EJ.f,(x,I) and a related free-boundary problem of Stefan type. Here d denotes tho Laplacean in the spatial variable x E IH”, h > 0, and the pair ix, /I’ are maximal monotone graphs in II? x Ft. If the first equation were to contain the term .‘ -kd(J(~, t)” with k > 0, then the system (I. 1) would be parabolic. The situation we consider here with k = 0 is accord,ingly a degenerate parabolic system, Although the system (1.1) with the (possibly multi-valued) nonlinear monotone graphs CY,,4 is of mathematical interest in its own right, we present in Section 2 an extensive discussion of how such a system arises as a model of heat conduction in a composite material consisting of two components in which a change of phase occurs in the second component. This model is 1 OOZZ-039+/83 Copyripbl .AlI ri~:llls EJ1983by ui‘ ruproc!ucriun $3.00 4ademic Press, Inc. in mv form rcseivcd 2 DIBENBDETTO AND SIIOWAL'I-ER described by (1.1) with ,8 obtained in the special form P(X) = h-u + .LH(x), where b > 0, L is the latent heat of fusion and H(.) is the multi-valued Heaviside step function. Certain models of diffusion through fractured porous media lead to the same system. Our results on (1.1) are organized as follows. In Section 3 we prove that the first initial-boundary-value problem for (I. 1) is well-posed when the data satisfy certain integrability conditions and ,8 is defined everywhere on II?. If the data are non-negative then the solution is likewise non-negative; this property is essential for the model problem discussed in Section 2. We make extensive use of the theory of maximal monotone operators in Hilbert space to which WCrefer to 12. 3 I. Certain properties of the solution arc obtained when we restrict attention to the case where ~1 has a lower linear bound and /I = b/ + LH as in the model problem. In Section 4 we show that if the data arc essentially bounded then the solution of (1. I ) is essentially bounded. Additional conditions on the data are shown to imply that 6, and 9 are continuous. In order to obtain these regularity results we found it very useful to treat the problem as an equation of evolution rather than to have formulated it as a variational inequality. In Section 5 we exhibit an explicit lower bound on the first component of the solution of (1.1). This implies that the set of points where this function is positive (the positivity set) is non-decreasing with time. Finally, we show that the positivity set of the first component contains that of the second component, and an example is given to show that this containment may be proper. 2. D~PFUSION IN HETEKOGENEOIIS MEDIA We begin with a mathematical description of diffusion processes within a medium consisting of two components. A fundamental assumption is that the first component occurs in small isolated parts that are suspended in the second component. This situation arises in thermal conduction through rocky soil, since the rocks arc isolated within the soil. It also occurs in lhe diffusion of liquid or gas through a porous media that has been fractured, since the blocks of the medium are isolated from one another by the system of fissures. Next we shall formulate a free-boundary problem of Stefan type that results from a change of phase in the second component of the medium. This arises in the model of heat conduction through the moisture in rocky soil since the soil moisture may freeze or thaw with a corresponding release of latent heat; thcrc is no moisture in the rocks. If WCconsider diffusion in a fractured medium in which the system of fissures is only partially saturated then we can think of the fissures as containing holes in which a certain A FREE-BOUNDARY PROBTXM 3 amount (per volume) of the liquid or gas is trapped and can no longer take part in the diffusion. Such a diffusion process is formally equivalent to the preceding heat conduction problem. Finally, we give a weak formulation of this one-phase Stefan problem for a two-component medium. Consider the conduction of heat through a heterogeneous medium G c. II:“> consisting of two components. As our first model for such a process we take the system of equations in the region 12 = C x (0, co), I (I $ - kAt9 + k(8 - w) =./;, where H and cp arc the temperatures in the first and second components, respectively. Each is a function of position x E G and time t > 0 and is obtained at a point x by averaging the temperature of the corresponding component in a neighborhood which contains a sufficiently large number of pieces of both components. The constants (I, b are specific heats of the respective components, k is the conductivity of the lirst, the conductivny of the second component is normalized to unity, an.d the positive number h is related to the surface area common to the two components. Thus h is a measure of the homogeneity of the material. The system (2.1) is just a pair of classical heat conduction equations together with a linear coupling to model the simplest cxchangc between components. Our basic assumption that the lirst component occurs in small parts isolated by the second component implies that k = 0 in (2.1). That is, the particles of the first. component may store heat (u > 0) or may exchange with the surrounding second component (h > 0), but they cannot pass heat directly to other firstcomponent particles (k = 0). This is the sense in which the system (2. E) is degenerafe ,uar-crbolic. Suppose there is a solid-liquid phase change in the second component at the temperature cp= 0. We consider here the (one-phase) situation whereln (n > 0 everywhere. The region fl is separated into a conducting region IL., where CJ> > 0 and a non-conducting region a,, where ip = 0; these correspond to completely melted and partially frozen parts, respectively. ‘WC need not assume that Q, consists exclusively of ice but only that it is a mixture of ice and water in thermal equilibrium at the melting temperature. At each point (x, t) of 12 we introduce the fraction of water, ((s, t); note that c E f-Q?) in R, where I!(.) is the maximal monotone Heaviside graph given by H(.s) == 1 for s > 0, f-f(O) = IO, 11. and M(s) = 0 for s ( 0. The two regions arc separated at time t by an interface S(l). If we let n be the unit normal on 4 Dl HENEDETTO AND SHOWALTER S(t) directed towards Q, and V be the speed of S(t) along n, then we obtain the condition av -=-LV(l-<) an on s(t), where i-iv/an = V, . n is the heat flux across S(t) and (1 - 5) is the fraction of ice. Moreover, if N = (N,, N, ,..., N,, N,) denotes the unit normal on the interface S = U { (S(l), t)}, we find that (2.2) is equivalent to V,a, . (N, ,..., NJ = LN,( 1 .-- r). (2.3) Each of (2.2) and (2.3) is called the interface or free-boundary condition. It is worthwhile to recall the simple experiment in which one applies a uniform heat source of intensity F to a unit volume of ice at temperature q = 0. The temperature remains at zero until L units of heat have been added. During this period there is a fraction < of water coexisting with the ice and 5 increases at a constant rate F/L. When all the ice has melted, r = 1 and the temperature (o begins to rise at the rate F/b. The constants L. and b arc the latent heat and specific heat, respectively. We can summarize the above by stating that the rate of increase of the internal energy or enthalpJ> ZI= bq + .Lr is given by F. Later we shall see that not only is enthalpy the natural variable to determine the state of the process but that it is mathematically the proper variable by which to describe the evolution of the process. We can now formulate our problem. With the notation above we seek a triple of non-negative real-valued functions I!?,w, l on Q which satisfy the following: r a g + h(6r - cp) =f, and t E H(v), in R, (2.5) b+lp+h(++S, in a+, (2.6) $+Lv(I-e;)=O on S, P-8 1 rp=o on aGx(O,co), (2.9) r A FREE-BOUNDARY PROBLEM 5 The data consist of the strictly positive numbers U, h, h, I, and the: nonnegative functions f,, Js on R and O,), qo, to on G For which we assume 5,,(x) E WV”C~)) f or all x E G. As before, we have set ~‘2, = {(x. t) E Lb : q>(?c,t) > 0) and R, = ((x, 1) E D : q~(x, t) = 0). The unknown intcrfacc S between 0, and a,, is the primary difficulty in the problem. It is approrpiate to obtain a weak formulation of the problem (2.4)--(2. i 0). This is necessary cvcn with smooth data because the free boundary S ma) vary in a discontinuous manner and it is also convenient because it casts the problem into the form of an evolution equation in Hilbert space. Thus we first compute i?u/iit ---&I in the sense of distributions on L?. For each te?,t function ~1E C;:‘(Q) WC obtain + !’ WP . (N, ,a”*,N,,,) + USC- 1) N,) V’. ‘S We have assumed that the interface S and the restrictions of (u and < to 0, and to s2, are sufficiently smooth to apply Gauss’ theorem. This calculation shows tha.t in P’(Q) if and only if (2.6), (2.7) and (2.8) hold. From these remarks we obtain the following weak or generalized formulation of the two-component Stefan problem: given T > 0 and the non-negative functions J, . ,f! on .(;! and 6 Dl BENEDETTO AND SHOWAI>TIIII f3,,, v,,,,,to on G with {,, E H(vo), find a non-negative triple of functions which satisfy &H ‘(0, T; L’(G)), (D=(bl+LH)-’ e(o) = 4, v, E L’(O, T; If;(G)), (u) and 40) = b+, -t a, t’ E H’(0, T; H-‘(G)), (2.1 1) in L*(O, T; L’(G)), (2.12) in f,*(O, T; H-‘(G)), (2.13) in L’(O, T; H,!,(G)), (2.14) in L*(G). (2.15) Certainly a smooth solution of (2.1 l)-(2.15) for which the level set S is a smooth manifold necessarily satisfies (2.4t(2.10). Remarks. The condition b > 0 arises later in the discussion of properties of solutions so we briefly indicate the significance of this assumption. The constant b is a measure of the storage capacity of the second component and it depends on the type of material and also the percentage present in the second component. Similarly, the constant L is determined by the type and percentage of this material in the second component of the medium. The essential interest here is in the change of phase phenomenon so we are concerned with the case of a sufficient percentage of the second component material being present to permit L > 0. The corresponding physically significant case is that of b > 0; otherwise we would be considering the unlikely case of a material with positive latent heat of fusion but with null heat capacity. Neventheless, most of our results to follow are obtained from the wcakcr assumptions that L > 0 and b > 0. The type of the problem we have called degenerate parabolic. In the system of partial differential equations (2.1) with k = 0 it is of interest to consider the case of b = 0 11, 7 J; one can then reduce it to the single partial differential equation which is of pseudo-parabolic type 16, 221. This is distinctly not the case for the free-boundary problem considered here. An elimination of 0 from (2.12), (2.13) leads to the evolution equation (alh)~+$(u+uyl-(nlh)dy?)-dy?=S,+f;+(a/h)~. (2.16) : A FREE-BOUNDARY PROBLEM The pairs of equations (2.14), (2.16) gives an equation for q which is of second order in time-derivatives, definitely not pseudo-parabolic unless both h = 0 and L = 0. Thus. even in the case of h = 0 where the local description of the problem contains a pseudo-parabolic equation (cf. (2.4) and (2.6) in rZ,), the fret-boundary problem with I, > 0 is not of this type. Problems where the phase is dctcrmincd by the first component can be pseudoparabolic; see 110, 18 I. 3. EXISTENCE AND UNIQUENESS OF T‘W: WEAK SOLUIION We shall prove that the weak formulation of the Stcfan problem (2.1 I b(2.15) is well-posed. This will be achieved by showing that the problem corresponds to an evolution equation whose solutions are determined by a nonlinear semigroup of contractions and that the generator nf this semigroup is a subgradicnt operator. The existence and uniqueness of a gencralizcd solution of the Stcfan problem is contained in the following. TI IHC)I<I;M 1. Let u and /I be maximal monotone graphs on /I j x let j and lc be proper convex lower-semi-contirzuous functions subgrudients are given by +j = a ’ and 2li =/I’ I. tls,s~r~ze u,, E L l(G), “m”) E L ‘(Gj, v,, E L’(G) n H l(G)> ./; E L2(0, 7-t L2(G))? 11i and whose I<(v,,j E I, ’ (Gj, .f; E L yo, 7’; ti -- j(G)), and that the domain ?[,!I is equal to II-?.Then there exists a unique quadrup!e of’functions which sati@ u E .H’(O, T: L’(G)), v E H’(0, i’-; H ‘(C)j. 6,E L ‘(0, 7’; L’(G)), q E L’(O, T; H;,(Gj). $ -- dy? -t- h(p - 0) ==J1 in u E u(e), a.e. in n, (3.4) v(0) = v. a.e. in G. (3.5) there is a pair O,, E l.“(Gj, q. E H:,(G) f’br which u(0) = U”, (a) (3.1) l[ in addition 2’ E P(v) L..‘(O, 7’; H--‘(G)). (3.3) 8 DI BBNEDE’TTO AND SHOWALTER u(, E u(B,J and v, E /l(tp,,) a.e. in G, and ifs, E H’(0, 7’; L2(G)),f2 E H’(0, T, H-‘(G)), then + SE E .Lcn(O, T; L*(G)), 0 E L*‘(O, T; L*(G)), L”“(O, T; H-‘(G)), cpE L’“(0, T; H:,(G)). (b) If in addition a(O) 3 0, p(O) 3 0 und each of the functions f, , fi, u0 and v,, is non-negative, then each of u, v, 8 and q is non-negative. Prooj Let V be the product space L’(G) x HA(G) which has the dual V*=L*(G)xH ‘(G). DefineREY(V, V*) by NV) = J(. (4 u, - u*)(v, - v*) -t vu, . Vl!*J, u = lu, 3U?], L'= Iv,,u*] E v. Renorm V with the equivalent norm (&.4(u))‘! so that B : V+ V” is the corresponding Riesz isomorphism of the Hilbert space V onto its dual. Note that B is given in Q’(G) in the form B([u,,u21) = Ih(u,- uz>, h(u,- u,)-41, Iu,,&I E v. Next we consider the function J : V” + 114 U (+c.o } defined by 4~) = j, i (Au ,>+ W2)) if u,EL’,j(u,)EL’, 4.4 UZEL’nH-‘, EL’, otherwise, u = [u,, u2 ) E V’“. =+a3 From 13, pp. 115, 123 1 we find that J is a proper, convex and lower-semicontinuous function on V”. Furthermore, the subgradient ofJ is determined as follows: g E iq.4) with g= Ig,,g,l and u = Iu, 3u*l in I/:‘: if and only if for some u = IL),, v21 E V we have g = B(v) and VI E au,>? v2 E ak(u,) a.e. in G. These computations are immediate from the corresponding results of 13] on the components of V”. A FREE-BOUNDARY 9 PROBLEM It is useful to characterize U explicitly as a composition of operators in Q”(G). Thus we define A : V+ V* by A = ICY]: that is, u EA(o) with U- Iu,.u~] E p and z’ = [ 21’, 1!21E C’ if and only if U, E a(~,) and U? E /?(u,) a.e. in 6. (Note that A .- ’ is the subgradient of J computed from the Banach space V* to its dual Vi:* == k and YI is the corresponding subgradient of the conjugate of J ( 11 I. j From t,hc computations above WC have the representation i;J = B 0 R --’ as desired. Given the subgradient operator t/J on the Hilbert space P, it is well known 12, 31 that the initial-value problem qp $- ijJ(w(t)) a.e. -f(t), 1 E. IO, T1, w(0) = I$‘() has a unique SE L’(0, T; Pi;) J’E. H’(0, I’; V”) remarks, with SW == I.f;(t),.f:!(r) (3.6) solution IV E H’(0, T, V’) whenever w,) E dam(J) and are given. Furthermore, if HJ(,E dom(&/) and then this solution satisfies dw/dt E I;“-(& T: Ye). These the identifications iv(t) = lu(t), U(Z)/ E V*, M’,,= Iz;(,, cc,/J and /O(l), q(l)1 = B -‘(j-(f) - w’(r)) E A ‘(w(t)), show that (3.6) is equivalent to (3.1 k(3.5 j and thereby establish all but (b) of Theorem 1. For the proof of (b) we first change the data as follows: (I) Set j(s) =,i(O) for s < 0 and leave the values as originally given for s > 0; thus dam(a) c IO, +a). (ii) Add to p(s) the quantity s for those s < 0 and leave the values as originally given for s > 0; thus B is strictly monotone on ( .--co, 0 1. Since ug is non-negative the hypotheses of Theorem I still hold so there is exactly one solution of (3.1b(3.5) with the modified data; WCdenote it by U, c, 0, w as before. Since the domain of u contains only non-negative numbers, it follows that 0 > 0. Our plan is to show that the remaining three functions are non-negative. Next we consider Eq. (3.3) written with right side M’-t.f, and initial condition ZJ~being non-negative. This equation is of independent. interest. LEMMA P. Let A = h -A be the indicled Kiesz map of’ the IIilbert space H:(G) nto its dual, H ‘(G), und let H ‘(6) hat!e the scalur-product corresponding to A. Lel y be u maximal monotone graph on 1-tx IFi which conlnins the origin und whose range is all of IF?. (a) The operator A 0 y is maximal monotone on W’(G) with rhe 10 domain DI RENEDETTO (c E H--’ f-l L’: AND SHOWAI,TER there is u ~1E H:,(G) with q(x) E y(v(x)) ax. XEG}. (b) 1 > 0. y’C= (JEH--‘(x):f>O}, then [I-tk4 oy] ‘(C)c-Cforevcry Prooj: Part (a) follows from Theorem 17 of 131, where it is shown that A o y is a subgradient on H- ‘(G). To verify (b), let (I -,i-AA 0 v)(v) =f in H ‘(G) with J> 0. By truncation and regularization we obtain a sequence j;, E L’(G)n C”(G)n C with S,,-J’ in H--‘(G). Since A 0 y is maximal monotone, the corresponding sequence u,! = [I t AA o 1~1 ‘fl converges to 21 in H-‘(G). From Proposition 5 of 151 it follows that each II,, E C, so WC have u E C. Let F be a maximal monotone operator on a Hilbert space H; let v,, E dam(F) and fE L’(0, 1’; H). Then there exists a unique Ice& solution of the initial-value problem 12, p. 64 ] f + F(v) 3f on 10,TI, v(0) = V”. By a weak solution we mean a uniform limit of strong solutions L’,, corresponding to data vi and f,, with v G+ vg and j;, --$fin H and L ‘(0, T, Ii), respectively. This existence result is proved by choosing the sequences above with each VI; E dam(F) and each f, a step-function with values from the range of .j- 12, p. 65 1. LEMMA 2. Let C be a closed cone in H. Ij’ tiOE C, f’(t) E C jtir all tE IO,?‘], und if[l+AF]-‘(C)cCj br all A.> 0, then the weak solution ~1 (?f (3.7) satisjies t)(t) E C jbr all t E (0, T]. ProoJ By the preckding remarks it suffices to consider the case of v,, E dam(F) and a step-function f given on a partition 0 = a, < .a. < a, = T by f =yi E C on [a,-, , ai). The solution is given inductively by v(0) = vu and v(f) = Si(t - aJ ~(a,) on [ai-, , ui], where Si is the semigroup generated -(F - yi). By 12, Proposition 4.5 ] it suffices to show by [T + A.(I; - yi)l ’ (C) c C, for then we have v(t) E C for all I E 10, r1. Thus, let x= II+A(F-yi)]-‘4 7 with y E C and A > 0. It follows directly that x = (f + A.F)-’ (Ayi + y). Since AJ!~+ y E C we have x E C and we are done. To obtain II > 0 in (3.3) we apply Lemma 1 with 1’=/I ’ and then apply Lemma 2 with F =A o y. Since /? is strictly monotone on (-a, 01 it follows from (3.4) that rp > 0. Finally, writing u as the sum of its positive and negative parts, u = u ’ - U-, we obtain, from (3.2), A FKEE-ROUNDARY i. I PROBLEM Since u and 19have the same sign and /ZCJJ +J; is nonnegative, the right side is non-positive so u- = 0. Thus all four of U, L’, 0, q are non-negative. II. follows that this quadruple is a solution of the original problem without the modified data. By uniqueness this is the solution of the original problem and (b) is established. Remarks. The essential point in the first part of the proof of Theorem i is to reduce the problem to the evolution equation (3.6) whose solutjon is kiac pair 1u(r), o(t) 1 of “enthalpy” functions associated with the weak solution. St is this sense in which enthalpy is the natural variahle for the problem. For the special case of.f? E L’(O, 7’;,L2(G)) we can give an alternate proof of part (b) of Theorem 1 as follows. Approximate /3 by a smooth fi, for which the corresponding solutions [U,,, on] can be shown to be non-negative by direct L’-estimates on (3.2) and (3.3). Then using methods of 191 we can let II --t co to obtain the non-negativity of Iu, c 1. Iiowevcr, the proof given above permits the more general data of the existence result, and we also obtain the corresponding the corresponding w&known non.-negativity result for the abstract porous media equation in H -j(G), where h > 0 and 7 is maximal monotone. We could not find this result in the literature. The evolution equation (3.6) is of the form B ’ is positive self-adjoint and A.-’ is maximal monotone from a Hilbert space to its dual. Various generalizations and related equations have been discussed in 14, 6, 9, 10, 15, 16, 20, 21 1. wllcrc 4. BOUNDEDNI’SS AND CONTINUITY OF THE Wt:.?r~ SOLLJTWN We shall prove that the “temperatures” tl and CJIin the weak solution are bounded when the data in the problem are bounded. WC also give sufficient conditions for 0 and C+J to be continuous. These results arc obtained in the following special case of Theorem 1 which contains the weak formulation of the one-phase two-component Stcfan problem. ‘rIIEOREM lhe following: 2. In addition to the conditions qf ‘I%rorem I(b) M’IZ~SSLMC 12 DI BENEDETTO AND SHOWALTER (i) (ii) There is a number a > 0 such that r > as for all r E a(s); (iii) the initial data and forcing terms are essentially bounded: Us,, and.f, ,.f, E L “(0). Then the functions u, v, I!?,v are bounded on the maximal monotone /I is of the form ,9 = bI + LH, where I is the identity, H is the Heaviside graph and both b and L are non-negative; v,, E L”‘(G) R. (a) If in addition we have (rl - r,l > a Is, --’ s2/ for all r, E a(~,) and rz E a(q), and if the functions u0 and lif,(., t) dt are un[formly (H6lder) continuous on G, then 8 is uniformly (respective!y. Hiilder) continuous on 0. (b) J7 b > 0 and q,, is uniformly un[formly continuous on D. continuous on G. then cp is Let u, v, 8, ~0be the solution of (3. I )-(3.5); by assumption (ii) we Prooj may write ZI= brp -I- L<, <E H(p), in a. For each c > 0 we consider the Steklov averages where q and 19are extended as y(O), e(O), respectively, on (--6, 0). It is known that lim,.+, qC= q (etc.) in L’(0, T, H:(G)) 114,p. 85 1.By integrating (3.3) over [t _- C,t 1 we obtain = ho,(t) + (l/c) [’ f;(s) ds. “f--C We shall apply this to (q(f) - k)’ , integrate over (0, f) x G where the superscript plus denotes the positive part of the indicated function in H:,(G) and the number k is chosen by k = Max Ilq+llr,d~cG,; + II~~11~~~~~) + $ (IIf, lIr.,ha,rjj + 2 IlhlLd( - and take the limit as c 1 0. To this end we obtain .I d(t)W) - k)+ = i IlW) - k) ’ /I:,w;)-- t lib+,-..-k) ’ ll:w;, 0 ci limR--0JJ and the last term vanishes since k > )/v. l)r,2(C;j: (r(t) - at - e)NP(t) - k) + > (T(t) -- lNPW - w ‘- = 0 A FREE-BOUNDARY since k > 0 and T(r) E H(q(fj); PROBLEM and _. f’ [ (h I(u, -- k)’ 1’ $- h&p-k) -0-c; ’ +- lV(ci, .-- k) 1’)” Thus we obtain ‘This leads immediately to the estimate k j’ j (q - k) -‘- t j; I,; /(p -- k) ’ /’ 0 c; Next we estimate the first term on the right side of (4.1). integrate (3.2) over (0, I) and use (i) and 0 > 0 to obtain (4.2,) From (4.2) it easily follows that NOW apply this to (q~--. k) ’ and integrate to obtain .I . 14 Dl BENEDETTO AND SHOWALTER Note that so we have from above j’ j O(cp -- k)’ &J: 0G II@- kc+Il:‘(G, If we use (4.3) in (4.1) we obtain Thus, if 0 < I < a/2h, then by our choice of k the right side is non-positive and the left side is necessarily zero. In summary, we have shown that with k as given above we have for a.c. I E [0, a/2h I. The first is immediate from our preceding calculations, the second follows from (4.2) and the third is obtained from (4.2) and Gronwall’s inequality. From the dependence of k on the data it is clear that the estimates (4.4) on G x (0. u/2h) can be extended to give a bound on u, 0 and q on all of Q in a finite number of steps. In order to prove (a) we first consider the functions @,(x,0 = j-i rp(x,s) ds, 0(x, 2)= j' 0(.x,s) ds. 0 A FREE-BOKJNDARY PROHLEM Integrate (3.3) to see that @ is a weak solution of Tf h > 0 then from 114, Theorem 1.1, p” 419 / wc conclude that @ is uniformly Hdlder continuous on 0. If b = 0 then from 113, Theorem 14.1, p. 201 1 WC conclude that @ is uniformly Hijlder continuous on G, unifurmi!; in t E 10, 7’1. Since cpis bounded, Q, is trivially Lipschitz in t. Thus it foiknvs that @ is uniformly HGlder continuous on R. Next we integrate (3.2) to get u(x, I) + j; h(O(x, s) -- cp(x, s)) ds -= f’/;(x: -0 s) ds -I- u,,(x). By taking the difference of this identity at x = .Y, , .x2 E G we obta.in - @(x2.t - j”.f,(.Y,,s) ds -- U(,(XJ. . (I (45j Ry our assumptions in (a) the left side of (4.5) bounds the quantity a / B(x, , t) - 0(x7, t)l and the function has a modulus of continuity have a(.) which is independent of I E 10, 7’1, so we a I@, , t) -- 0(x, ) t)l < h 1’ I@, , s) - H(.x2, s)l ds + IS(X, -- x>)~ a0 By Gronwall’s inequality it follows that 19 has the same tr~od~~lus of continuity, cr. in X. From (3.2) follows the uniform Lipschitz continuity in i of u and then the assumption in (a) shows that 0 is uniformly Lipschitz in 1. This finishes the proof of (a). 16 DIBENEDETTO AND SHOWALTER The proof of (b) is an immediate corollary of [S, Theore 5.3, p. 691. The point is that q is an essentially bounded weak solution of with +/at p. 501 I. E L’(B). This last inclusion follows from b > 0; see [9 1 or 114, Remarks. The boundary aG of the region G is assumed to satisfy ‘Condition A” of 1131 in both (a) and (b) of Theorem 2. That is, there is a pair of positive numbers a, and 8, such that for any sphere B,. with center on i!G of radius Y < u,, and for any component 6, of the intersection G,. = B,n G it follows that mes(G,.) < (1 - 0,) mes(B,). Without such a restriction on the smootness of the boundary of G we obtain local or interior continuity results as above. It is not known whether cp is continuous in Q in the case h = 0. 5. ADDITIONAL PROPERTIES OF THE WEAK SOLUTION Under rather general conditions on the data in the weak formulation of our problem it follows that the positivity set of the enthalpy u is increasing with time. This is equal to the positivity set of the temperature ql. An example shows this containment may be proper. The preceding properties of the weak solution will be obtained in part from the following comparison result. LEMMA 3. Suppose t, < I’ and for each 1E It,,, TI we are given a pair y,(t, a), y,(t, a) oJgraphs on R X r? such that y2(t, .) is monotone and for each s, E y,(t, r) there exists an s2E yz(t, r) for which s,<s,. (5.1) Let the pair of absolutely continuous functions u, , u2 : It,,, T] + IIi satisjj~ ul(b) G u2(~d and for a.e. 1E It,,, T]. Then u,(t) < u2(t) for I, < t < T. ProoJ: Suppose there is a t2 E (t,, T] such that u&&> u2(t2). Define I, = lubjl E [t,, T] : u,(t) < u,(t)} and note that u,(t,) = u,(t,) and u,(t) > u2(t) for all I E It,, t,l. For each t E [li, 1,1 for which -u:(t) E y,(t, u,(t)) A FREE-BOUNDARY i 7 PRODI.EM there is by (5.1) an s(f) E I’~(& U,(I)) with s(l) < ---u;(t). For SLICII a E WC obtain 04(l) - G(t>>tu,(t)- u,(t)) < (44 -- M)>(~~,@)- U*!f)) and this last quantity is non-positive because ~~(1,.) is monotone. Thus the function (u,(tj -- u,(t))* has a non-positive derivative on [r,. f2 1. it therefore vanishes on It,, f?] and this contradicts the choice of tz. Suppose we are in the situation of Theorem l(b). For a.e. x’ E G the function uz(l) E U(X, t) is an absolutely continuous solution of (3.2). In order to apply Lemma 3 to (3.2), (3.4), define )qt, 11)= ha -I(u) ---hrp(x.1) --./;(x,rj. y,(l? 24)= (h/a) u + -ji(X, t), WC will assume u > 0 and that for each r > 0 there exists an s E LT ‘(Y) such that US < r. (5.2) This implies that (5.1) holds for our choice of y,, y:. L.et U, be the solution of g(t) + (h/a) u,(t) =J;(A 0, with ~,(t,,) = U(X, t,,). From following. 0 < t ,< 7; Lemma 3 we obtain the first part of the THEOREM3. Jtz addition to the conditions of 7’heorern J(b) hl!e MSLU~~ there is UN a > 0 j’or which (5.2) holds. Therz the ,first component q,f’ the solulio~z qf’ (3.1 t(3.5) satiA$es qx, 1) > e- (h:o)lr- f,,l , u(x, t,,) i- if “‘fi(,.c, s) ds, e-- I!~.‘~l~~f-10 0 < t,, < t < 7; (5.3) for almost every x E G. Thus the set S,?(u) E (x E G : u(x, E) > O} is increasing with t. Furthermore? the set S ‘(u) z {(x, t) E Q : u(x, t) > 0 j contains the interior of S ’ (f,) and u{S~-(ql):O<t<l,}C1S,~(U), O<l,<T. (S.4) Prooj: The inequality (5.3) follows from the preceding remarks and it immediately implies the monotonicity of Si (u) and the inclusion of the interior of S ‘-(s,) in S’(U). We verify (5.4). Let x1 G?S,‘,(U), that is.. for all O< t<:~,. ‘Thus u(xI, t,)=:O, so by (5.3) we have u(x,, t)=O iiu(x,, /)/at = 0 for 0 < t < 1,. From (5.2) we obtain (see below) 0(x,, t) =: 0 18 DIBENEDETTO for 0 < I < t, so (3.2) implies &, for all 0 < t < 1,. COROLLARY. Proof. AND SHOWALTEH , t) = 0 for 0 < t < t,. That is, x, & S:(q) In the situation qf Theorem 2(b) M’e have S+(rpj c S ’ (u). Since v, is continuous this follows from (5.4). Remarks. The condition (5.2) is actually equivalent to the assumption (i) in Theorem 2: r > as for all (s, r) E 0~.To see this, note that if (so, r,)) E u with r,, < as,, then we can choose r, = (1/2)(r0 + as,J > 0 and from (5.2) a s, with (s,,r,)Ea and as,<r,. But then s, < (1/2)(r,,/a + s,J < sg and contradicting the monotonicity of LT.Thus (5.2) implies that. all of (1 rl > ro, lies above the graph of r = as. As a consequence of the above remark it follows from (3.4) that U(X, t) > a@, I), hence S l.(0) c S+(U). If in addition we have a(O) = (O}, then s k (fq = s+ (2.4). In the case of our original problem, (2.1 lb(2.15), we have S ’ (q) c S ’ (8): Thus 6, > 0 in the region Q+ where the water is completely melted. The following example shows that we do not necessarily have s+((o)=s’(e). Define B(t) = em.‘, a, = 0 and v(f) = 1 - em-’ for t > 0. This EXAMPLE. triple of non-negative functions is the solution of (2.11) - (2.15) with L=a=h= 1 and arbitrary b>O,f,=f,=O, and BO= I, ~o=<o=O. 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