1.    Stefan Problem with Dirichlet Boundary Condition

In a domain Па = {(t, x) € R2 : t > 0, x > ay/t} we consider the multi-phase Stefan problem for the heat equation ut = a2uxx,  Ui

• ^#Supported by the Ministry of Science and Higher Education of the Russian Federation in the framework of the State task.

where uo < ui < ••• < um < um+i = ud, Ui, i = 1,...,m, being the temperatures of phase transitions, ai > 0, i = 0,... ,m, are the diffusivity constants. We will study continuous piecewise smooth solutions u = u(t,x) in Па satisfying (1.1) in the classical sense in the domains ui < u(t,x) < ui+i, i = 0,...,m, filled with the phases. On the unknown lines x = Xi(t) of phase transitions where u = ui the following Stefan condition dix'i(t) + kiux(t,xi(t)+) - ki—iux(t,xi(t)-) = 0                     (1.2)

is postulated, where k i >  0 is the thermal conductivity of the i-th phase, while d i ^ 0 is the Stefan number (the latent specific heat) for the i -th phase transition. In (1.2) the unilateral limits u _x (t, X i (t)+), u _x (t, X i (t)-) on the line x = X i (t) are taken from the domain corresponding to the warmer/colder phase, respectively. For physical reasons, the Stefan numbers d i should be positive. We will study an even more general case d i ^ 0.

In this case the problem (1.1), (1.2) is well-posed for u o C u C u d and reduces to a degenerate nonlinear diffusion equation (see [1], [2, Chapter 5])

g(u) t - K(u) xx = 0,                                   (1.3)

where g(u), K(u) are strictly increasing functions on [u o , u d ] linear on each interval (u i ,u i+i ), i = 0,... ,m, with slopes K ‘ (u) = k i , g'(u) = k i /a ² , and such that

K (u i +) - K (u i - ) = 0, g(u i +) - g(u i -) = d i , i = 1,...,m.

We will study the initial-boundary value problem with constant initial and Dirichlet boundary data

u(0, x) = u o , x >  0;    u(t, ay/t) = u d , t >  0.                    (1.4)

By the invariance of our problem under the transformation group (t,x) ^ (A ² t, Ax), A G R, A > 0, it is natural to seek a self-similar solution of problem (1.1), (1.2), (1.4), which has the form u(t,x) = u(—), £ = x/y/t. In view of (1.4),

u(a) = u d , u(+ w ) = lim u(—) = u o <  u d .

ξ → + ∞

Thus, it is natural to suppose that the function u(—) decreases. The case when u d C u o can be treated similarly. Certainly, in this case the function u(—) should be increasing.

For the heat equation ut = a2uxx a self-similar solution must satisfy the linear ODE a2u" = -—u'/2, the general solution of which is u = CiF

ξa

+ C 2 ,

C _i , C ₂ = const,

where F (—) =

ξ

• 1    S

e 4 ds

• 2    7П J

  -∞

This allows to write our solution in the form

^{u ( — )} = u i +-- ^{u +} - uTTy f^Ff~^- F^—'] \ ^£ i +i <  ^— < ^—i ,i = ^{0 ,} ... ^,m , (1-5)

F ξi+1 - F ξi        ai          ai ai             ai where +w = —o > —i > • • • > —m > —m+i = a and we agree that

F (+to) =

The parabolas £ = £ i , i = 1,... ,m, where u = U i , are free boundaries, which must be determined by conditions (1.2). In the variable ξ these conditions have the form (cf. [3, Chapter XI])

d i £ i     k i ^{( u} i+i ^- u i ^{) F} ‘ ( a i )      ^k i - i ^{( u} i ^- u i - i )F ‘ (a^)

(1.6)

+

• ² a . ( F ( ' ) — F (§))    a . - i ( F (ofc) — F (fe))

In the case m = 1 system (2.1) reduces to a single equation, which can be easily analysed. As a result, we obtain the classical Neumann solution of the Stefan problem. To investigate the nonlinear system (2.1) in general case of arbitrary number m of phase transitions, we notice that this system is a gradient one and coincides with the equality V E(£) = 0, where the function

_

£ = (£ i ,...,£ m ) € Q, (1.7)

E(O = ^— E ^k i (“ i + ⁱ - “ i ^)ln (F(|) ^— г(^о;))+ E df i- , i=0                      '   '   ⁷         '      ^{7 7} i=1

the open convex domain Q C R ^m is given by the inequalities £ i >  • • • > £ _m > a. Observe that E(£) € C “ (Q). Since the function F (£) takes values in the interval (0,1), all the terms in the first sum of expression (1.7) are positive. Therefore, E(£) > 0.

• 1.1.    Coercivity of the function E and existence of a solution. We will call a function (or a functional) ^ coercive if it tends to + w when the argument approaches the extremes of the domain on which it is defined. This may be exactly formulated as compactness of the sub-level sets { ^ £ c } . Let us introduce the sub-level sets of the function E(£)

Q(c) = { £ g Q : E ( £ ) £ c } , c> 0.

Proposition 1 (coercivity) . The sets Q(c) are compact for each c > 0. In particular, the function E(£) reaches its minimal value.

< If £ = (£ i , ■■■ ,£ m ) € Q(c) then

-

k i (u i+i

^— ш ^— F^) )^{£ E ( f ) £} c

i = 0,... ,m.         (1.8)

It follows from (1.8) with i = m that F(£m/am) — F(a/am) ^ e c/(km(um+i um)), which implies the low bound

£ m >  r i = a m F ^-i (F^)

- ^c

+ e km ⁽ um +i ^- um )

)

>α.

Similarly, we derive from (1.8) with i = 0 that

1 —

F a0

- ^c

^ e k ot u i -u o)

(notice that F (£ o /a o ) = F (+ot) = 1). Therefore, F (£ _i /a o ) £ 1 — e ^{- c/ ( k o (} u i ^{- u o ))} . This implies the upper bound £ _i £ r ₂ = a o F ^-i (1 — e ^{- c/ ( k o ( u} i ^- u ^{o ))} ).

Further, it follows from (1.8) that for all i = 1,... ,m — 1

f/^ — FfM a i             a i

- ^c

^ S i = e Mui +x -ui )

> 0 .

(1.9)

Since F ‘ (£) = 1/(2 ^ n )e ^ 2 /4 <  1, the function F (£) is Lipschitz with constant 1, and it follows from (1.9) that

K = { £ = (£ 1 , ...,£ m ) G R m : Г 2 >  £ 1 > ••• >  £ m >  r i , &  — Ui >  a i ^ i ( V i = 1,...,m — 1) } .

Obviously, K C Q. Since E(&) is continuous on K , the set Q(c) is a closed subset of K and therefore is compact. For c > N = inf E(£), this set is not empty and the function E(£) reaches on it a minimal value, which is evidently equal to N . >

We have established the existence of minimal value E(£ o ) = min E(£). At the point £ o the required condition V E(£ o ) =0 is satisfied, and £ o is a solution of the system (2.1). The coordinates of ξ 0 determine the solution (1.5) of our Stefan problem. Thus, we have established the following existence result.

Theorem 1. There exists a self-similar solution (1.5) of the problem (1.1) , (1.2) , (1.4) .

• 1.2.    Convexity of the function E and uniqueness of a solution. In this section we prove that the function E(£) is strictly convex. Since a strictly convex function can have at most one critical point (and it is necessarily a global minimum), the system (2.1) has at most one solution, that is, a self-similar solution (1.5) of the problem (1.1), (1.2), (1.4) is unique. We will need the following simple lemma proven in [4] (see also [5, 6]). For the sake of completeness we provide it with the proof.

Lemma 1. The function P (x, y) = — ln(F(x) — F(y)) is strictly convex in the half-plane x > y.

• < 1 The function P (x, y) is infinitely differentiable in the domain x > y. To prove the lemma, we need to establish that the Hessian D ² P is positive definite at every point. By the direct computation we find

d ² _p, . ₌ (F ‘ (x)) ² — F ‘‘ (x)(F(x) — F (y))

dx ² P⁽x ^{,y )}            (F(x) — F (y)) ²          ’

_d_            (F ‘ (y)) 2 — F ‘‘ (y)(F (y) — F (x))     _d_                F ‘ (x)F ‘ (y)

dy ² P ^{( x,y )}             (F (x) — F (y)) ²         ’    dxdyP ^{( x,y )}      (F (x) — F (y)) ² ‘

We have to prove positive definiteness of the matrix Q = (F (x) — F (y)) ² D 2 P (x,y) with the components

Q ii = (F ‘ (x)) ² — F ‘‘ (x)(F (x) — F (y)),

Q 22 = (F ‘ (y)) ² — F ‘‘ (y)(F (y) — F (x)),   Q i2 = Q 2i = —F ‘ (x)F ‘ (y).

Since F ‘ (x) = 1/(2 ^ n )e ^{- x} 2 ^{/ 4} , then F ‘‘ (x) = — (x/2)F ‘ (x) and the diagonal elements of this matrix can be written in the form

Q ii = F ‘ (x) ( x2(F (x) — F (y))+F ‘ (x) ) = F ‘ (x) ( |(F (x) — F (y))+(F ‘ (x) — F ‘ (y)) ) +F ‘ (x)F ‘ (y), Q 22 = F ‘ (y) ( 2 (F(y) — F(x)) + (F ‘ (y) — F ‘ (x)) ) + F ‘ (x)F ‘ (y).

By Cauchy mean value theorem there exists such a value z G (y, x) that

F ‘ (x) — F ‘ (y) _ F ‘‘ (z) _ z

F (x) — F (y) = F ‘ (z) = — 2 •

Therefore,

Q ii = F ‘ (x) (FW-^yMx - z) + F ‘ _{( l )} F ‘ (y),

Q22 = F'(y) IFixl—FWL-y’ + F'(x)F‘(y), and it follows that Q = Ri + F‘(x)F‘(y)R%, where Ri is a diagonal matrix with the positive diagonal elements

F‘(x)(F(x) - F(y))(x - z)    F‘(y)(F(x) - F(y))(z - y-    while R2 = Q1  11) .

2                  ’                      2

Since R i > 0, R 2 ^ 0, then the matrix Q >  0, as was to be proved. >

Remark 1. In addition to Lemma 1 we observe that the functions P(x, —от ) = — In F (x), P (+ot ,x ) = — ln(1 — F(x)) of single variable x are strictly convex on R. In fact, it follows from Lemma 1 in the limit as x > y ^ —от that the function P(x, —от ) = lim y , ^ P(x, y) is convex on R. This implies that

(F(x)) 2 d^ P(x, —от ) = ''    [ F ‘ (x) ( 2 (F (x) — F(y)) + (F ‘ (x) — F ' (y)) ) + F ‘ (x)F ‘ (y) ]

= F ‘ (x) ( x2 f (x) + F ‘ (x) ) >  0.

If (d ² /dx ² )P(x, —от ) =0 at some point x = x o G R then (x o /2)F(x o ) + F ’ (x o ) = 0 and therefore, x o is a minimum point of the nonnegative function (x/2)F (x) + F ’ (x). Hence,

0 = (x F(x) + F'(xd'x) = F^ + F''(xo) + F’(xO)xo = F^ > 0, x 2                  /              2                         2             2

which is impossible. Hence, (d ² /dx ² )P(x, —от ) >  0 for all x G R and the function P(x, —от ) is strictly convex. The proof of the strict convexity of P (+ot ,x ) is similar. In the limit as x < y ^ +ot we derive from Lemma 1 that the function P (+ot ,x ) = lim y , . ^ P(y,x) is convex on R and in particular

(1 — F (x)) ² dx P (+ot ,x )= 'i + [F ' (x) (|(F (x) — F (y)) + (F ' (x) — F ' (y)))+F ' (y)F ' (x)] = F ' (x) ( |(F(x) — 1) + F ' (x) ) >  0.

Assuming that (d‘ ² /dx‘²')P (+ot ,x ) = 0 at some point x = x o G R, we find that ( x o / 2)( F ( x o ) — 1) + F ^' ( x o ) = 0, which implies that x o is a minimum point of the nonnegative function ( x/ 2)( F ( x ) — 1) + F ^' ( x ). Therefore,

0 = ( x(F(x) — 1) + F ' (x) ) ' (x o ) = F(^   ¹ + F '' (x o ) + Fix^ = F ^{( x} )  ¹ < 0.

This contradiction implies that ( d ² /dx ² ) P (+ от , x ) >  0 for all x G R and, therefore, the function P (+ot ,x ) is strictly convex.

Now we are ready to prove the expected convexity of E(f).

Proposition 2. The function E(^ ) is strictly convex on Q .

• < 1 We introduce the functions

  E K) =

  
  

  _

  
  

  k i (u i+i

  
  

  _

  
  

  u i

  
  

  ^)ln (^F(ai)^— F ))’

  
  

  i = 0,..., m.

  
  

By Lemma 1 and Remark 1 all these functions are convex. Since m. .<2

E (0 = t>( « ) + E c ( ! ) + E 4 i =1                      i =1

and all functions in this sum are convex, it is sufficient to prove strong convexity of the sum

m

Ed) = E ®(« )• i=1

By Lemma 1 and Remark 1 all terms in this sum are convex functions. Therefore, the function E is convex as well. To prove the strict convexity, we assume that for some vector z = (Z 1 ,...,Z m ) € R m .

d ² E ⁽ f ⁾

^DE^^z • ^z = E         ^z j =0-               (^O)

i,j=1   ’     '

Since

m

0 = D²E(f ) Z • z = Ed²e< ) z • z i =1

while all the terms are nonnegative, we conclude that

D^E-id ) z • z = 0, i = 1,...,m.                        (1.11)

By Lemma 1 for i = 1,..., m — 1 the function E i (f) is strictly convex as a function of two variables f i , f i ₊i and it follows from (1.11) that z i = z i+1 =0, i = 1,... ,m — 1. Observe that in the case m = 1 there are no such i . In this case we apply (1.11) for i = m . Taking into account Remark 1, we find that E _m (f ) is a strictly convex function of the single variable f _m , and it follows from (1.11) that z _m =0. In any case we obtain that the vector z = 0. Thus, relation (1.10) can hold only for zero z , that is, the matrix D ² E(f) is (strictly) positive definite, and the function E (f) is strictly convex. This completes the proof. >

Propositions 1, 2 imply the main result of this section.

Theorem 2. There exists a unique self-similar solution (1.5) of problem (1.1) , (1.2) , (1.4) , and it corresponds to the minimum of strictly convex and coercive function (1.7) .

Corollary 1. The phase transition parameters f i , i = 1,...,m , corresponding to a solution (1.5) of problem (1.1) , (1.2) , (1.4) depend continuously on the Dirichlet data u D .

• <    Since the function E (f) is strictly convex and continuous with respect to the parameter U _m +i = u d , its minimum point f = f ( u d ) is a continuous function of this parameter. >

As follows from Corollary 1, the unique solution (1.5) depends continuously (in the uniform norm) on the Dirichlet data. Let us demonstrate that this dependence is also monotone.

Corollary 2. Let u _r = u _r (f) be solutions (1.5) of problem (1.1) , (1.2) , (1.4) with Dirichlet data U Dr , r = 1, 2 . If u di ^ U D2 then u i (f) ^ U 2 (f) for all > > ct.

• <    Assuming the contrary, we find a point в > a, where ui(e) < и2(в). Since ui(a) = udi ^ ud2 = U2(a) and the functions ui,U2 are continuous on the segment [a, в], there exists such a’ € [a, в) that Ui(a‘) = U2(a‘). Then ur(f), f > a’, r = 1, 2, are different solutions (1.5) of the same problem (1.1), (1.2), (1.4) in the domain П_а‘, which contradicts to the uniqueness statement of Theorem 2. >

2. Stefan Problem with Neumann Boundary Condition

Remark 2. In paper [5] Stefan problem (1.1), (1.2) was studied in the half-plane t >  0, x ∈ R with Riemann initial condition

u(0,x) = / “ + ’ x> 0 ^,                             (1.12)

I u - , x <  0.

Actually, this problem coincides with the problem (1.1), (1.2), (1.4), if we take a = —от . Solutions of problem (1.1), (1.2), (1.12) have the same structure as in (1.5) and correspond to a unique minimum point of the function similar to (1.7) with only the difference that the parameters ξ i can take arbitrary real values. In this section we mainly follow the scheme of paper [5], see also the paper [6], devoted to the case a = 0 of the fixed ray.

In paper [4] the Stefan–Riemann problem (1.1), (1.2), (1.12) was studied in the case of arbitrary (possibly negative) latent specific heats d i . We found a necessary and sufficient condition for coercivity of Efy, as well as a stronger sufficient condition of its strict convexity. The similar results can be obtained for the Stefan–Dirichlet problem (1.1), (1.2), (1.4).

Also remark that in recent paper [7] a variational description was given for self-similar entropy solutions of nonlinear convection-diffusion equations (with the Dafermos diffusion). It is shown that the corresponding variational functional (i. e., the potential) can now take its minimal value at boundary points, that leads to inequalities in the criterion of minimality. We demonstrate in [7] that these inequalities coincide with the known Oleinik–Carrillo entropy conditions.

Now we consider the Stefan problem (1.1), (1.2) in П _а , where we suppose now that a ^ 0, with the constant initial data u o and with Neumann boundary condition:

u(0,x) = u o , x >  0; K (u) _x (t,aVt ) = t - ² b N ,t> 0,              (2.1)

where K(u) is the diffusion function in equation (1.3), now defined on infinite interval [u o , + от ) (we take u m+i = + от ), and b N <  0 is a constant. The specific form of Neumann boundary data is connected with the requirement of invariance of our problem under the scaling transformations (t,x) ^ (X²t,Xx), X >  0. This allows to concentrate on the study of self-similar solutions u = u(x/y/t ) of the problem (1.1), (1.2), (2.1). For such solutions conditions (2.1) reduce to the requirements

K (u) ’ (a) = b N , u (+ot) = u o .                            (2.2)

Since K(u) is a strictly increasing function and b N <  0, we will assume that the function u(^) decreases. The case b N >  0 corresponds to an increasing u(^) and can be treated similarly. For homogeneous Neumann problem b N =0 there is only the constant solution u = u o . Let u i , i = 1,... ,m, be all phase transition temperatures in the interval (u o , +ot) so that u o < u i <  • • • < u m < u m+i = +ot. Assume that u = u(^) is a decreasing self-similar solution of (1.1), (1.2), (2.1). Then u(a) > u o and there is an integer n, 0 C n C m, such that u _n < u(a) C u n +i . We call this number n (i. e., the number of phase transitions) a type of solution u. The Neumann condition for a solution of type n reads k _n u'(a) = b N (notice that k n u x is exactly the heat flow through the boundary point). A solution of type 0 does not contain free boundaries and can be found from the formula

O = u o +        ■ ( f ( ) — 1 }                   (2.3)

k 0 F ^′ _a ^α ₀         a 0

As is easy to verify, kgu'(a) = bN, u(+to) = ug and requirement (2.2) is satisfied. By easy computation we find

"   “° + ab    (Ffe) - 0’ therefore, the necessary and sufficient condition for existence of a solution (2.3) is the inequality

(F0 - 0 * u ¹ ’

.    a0bN u0 + koF' (t)

which can be written in the form

^— b N * Y 1 =

Wui—MF^

M1 — F ( a )) ’

(2.4)

A solution of type n >  0 has structure similar to (1.5) (with m = n)

u(« = U i +           (fA — F А У

F ξi+1 - F ξi        ai ai

&+1 <£<£ i , i = 0,...,n — 1, (2.5)

<)= un + . F  (FA - FO “^^^

The necessary (but not sufficient, as we will soon realize) condition u(a) * U _n +1 of existence of such a solution has the form

u(a)=$ND (—bN )=un - ANny (F(an)— F(ft)) * u"«

(if n = m then it is always fulfilled since u m +i = +to). In (2.7) we introduce the Neumann-to-Dirichlet mapping, which maps a Neumann data —b N to the Dirichlet data u(a). We underline that the value ξ _n depends on - b N .

Assume that u(^) is a solution (2.5), (2.6) of type n. On a phase transition lines £ = £ i , the Stefan condition reads

d,«, ,   <“« ^{— U}i^)F' ft)     ,        ^{( u — U} i - 1 ^{) F} '(h)

^{2 +} ’ A(^) — F (&))’ ^{< - 1} ^ft(ah) — F(fe ))

. ₊ b N Fft) _— k      (U n — U- 1 )F'(A

^{2      F} ' ( an )      n ^-1        ( ah ) — ^F ( fei ))

i = 1,..., n — 1,

(2.8)

i = n. (2.9)

Like in the case of Dirichlet boundary condition, this system turns out to be gradient one, it coincides with the equality ^E _n = 0, where the function

n - 1

E n (0 = —E ki(Ui +1

i =0

-

u)h (FA—FA ))

,   ^a n b N

⁺ F ' () a n

F^ +1 E ^di& <=A,...,A € A, \ a_nJ  4^

4    '        i=1

(2.10)

Qn is an open convex set in Rn consisting of vectors with strictly decreasing coordinates lying in the interval (a, +to). Remark that F‘‘(s) = -s/2F‘(s) < 0 for all s > 0. Since bN < 0, this implies that the term fa^N ) F(fn/an) is a strictly convex function of the single variable fn on the interval [a, +to). In the same way as in the proof of Proposition 2 we find that the function n—1

En Ш = "E ki (ui+i i=0

-

u i

- f^        d i f ?

is strictly convex on a domain Q _n = {f = (f i ,..., f _n ) E n ( ^f ) = E n ( f ) + Fo ra y ^F () , ^{the function} E n ⁽f ’ an

€ R n : f i >  • • • > f _n ^ a } D Q _n . Since is strictly convex on Q _n as well. Let us

demonstrate that this function is coercive on Q _n .

Proposition 3. For all c € R the set Q _n (c) = { f € Q _n : E n (f) C c } is compact.

< Suppose that f € Q _n , E _n (f) C c. Then

E n а ) = — E k (ui +1 — u )ь (F(|) — F^))

+ ^{1 E} df ² = E ⁽ O - a nN^ ПУ C d = c - -an b N-

• 4 £1 i ' } F‘(at) UJ 1 F'(at)

Since all the terms of the left-hand side of this inequality are positive, we obtain the relations

- k i (u i +i

"-’*" (F(£) " F^))

C c i ,

i = 0,... , n — 1,

(2.11)

the same as inequalities (1.8). As follows from (2.11), the set Qn(c) = 0 if ci C 0. Therefore, we may (and will) suppose that ci > 0. Arguing as in the proof of Proposition 1, we derive from (2.11) the bounds fi C Г2 = aoF i (1

-

^c 1

e k o( u i ^-u o)

,

f i — f _{i +}1         — ^^1

----------- ^ $ = e ^ki ^{( u}i +^{1 ui )} > 0, i = 1,..., n — 1. a i

Thus, the set Q _n (c) is contained in a compact

K = {f = (f i ,--.,f n ) € R n : Г 2 >  f i > ••• >  f n >  a, f i — f i +i >  a i $ ( V i = 1,...,n — 1)}

(notice that this set is empty if r ? C a). Since E _n (f) is continuous on K C Q _n , the set Q _n (c) is a closed subset of K and therefore is compact. This completes the proof. >

It follows from Proposition 3 and the strict convexity of function E _n that there exists a point f ⁿ = (f n ,... , f n ) € Q _n of global minimum of E _n , and it is a unique local minimum of this function. There are two possible cases:

• A)    f ⁿ € Q n , i. e. f n > a. If, in addition, condition (2.7) is satisfied then there exists a unique solution (2.5), (2.6) of type n with f i = f n , i = 1,..., n;

• B)    f ⁿ € Q n , i. e. f n = a. Then E n has no critical points in Q _n , and a solution of type n does not exist. Let us investigate this case more precisely. The necessary and sufficient conditions for the point f ⁿ = (f n , ...,f n _— i ,a) to be a minimum point of E _n (f) are the following

^E n ( f ⁿ ) =0, i = 1,...,n — 1,                     (2.12)

∂ξ i

0         \

(2.13)

dfn En (f ”) *0

where condition (2.12) appears only if n >  1. Notice that for such n

En(S1,..., Sn—1 ,a) = - £ ki (Ui+1 - Ui) In (f (ai) - F (^)) +4 £ di ^2 i=0                     '   \ i/       \   i//i=1

,  anbN  a a \ n dna2    л„

⁺     a a \ ^F\ I ⁺ /1   ,   ^ n — ^a, ⁽S 1 ,---, S n — 1 ) ^ ^ n — 1 .

F Ian?     an'

We see that E(S 1 ,..., S n - 1 ) — E _n (S i , ■••> S n - 1 , a) coincides (up to an additive constant) with function (1.7) corresponding to Stefan-Dirichlet problem (1.1), (1.2), (1.4), where m — n — 1 and u d — u _n . Relation (2.12) means that VE(S n , ..., S n - 1 ) — 0, that is, (S n ,..., S n _— 1 ) € ^ _n— 1 is a unique minimal point of E(S 1 ,..., S n - 1 ). According to Theorem 2, the coordinates S in , i — 1,...,n — 1, coincide with the phase transition parameters S i of the unique solution (1.5) of problem (1.1), (1.2), (1.4) with u d — u _n (in particular, they do not depend on the Neumann data b N and on parameters a i , k i , d i with i ^ n). As is easy to calculate, condition (2.13)

reads

,          , dn ^a       ,         . ^{k n — 1 (} u ^{n U n — 1 ) F} ‘ ( n - ij

~b_N * Y n ⁺ -v>^where Y n ^—--- —7^—----;----

²                 a n — 1 ( F (^) — F ( a n a - 1 ))

.

(2.14)

In the case n — 1, we find that S n _— 1 — S o — + ^ , so that F(S rr _— 1 /a _n— 1 ) — F (+ ^ ) — 1, and the constant Y 1 coincides with the value introduced in (2.4). Under requirement (2.14) the case B) is realised so that a solution of type n does not exist.

We introduce the Dirichlet-to-Neumann mapping Ф ^DN , which maps a value r € (u _n ,u n +1 ] to a value — k _n u ‘ (a), where u — u r (S) is a solution of the Stefan-Dirichlet problem (1.1), (1.2), (1.4) with m — n and u d — r. This mapping is inverse to the Neumann-to-Dirichlet mapping ^ N^D introduced in (2.7) above. Notice that u m +1 — + w and in the case n — m the interval (u _n , u n +1 ] is replaced by the unbounded interval (u _m , + ro ). It follows

from (1.5) that

^ D^N (r) —

k n F ‘( a n) ⁽r - u n ) a n (F ( a n ) — F ( a ))

(2.15)

where S n — S n (r) depends on the Dirichlet data r — u(a). Notice that F (S o /a o ) — F (+ ^ ) — 1 and

koF ‘ (— ) (r — uo)

^Ф(r^{) —} J4(<, u o

(2.16)

is a linear function. By the construction u r (S) is a solution of type n to the Stefan-Neumann problem (1.1), (1.2), (2.1) with Neumann data b N — —^ D^N (r). Therefore, condition (2.14) cannot hold, and we claim that Ф ^DN (r) > Y n + d n a/2 for all r € (u _n ,u n +1 ].

Lemma 2. The function & D^N (r) is strictly increasing continuous function on (u _n ,u _n +1 ], such that

^ DN ^{( u} n ^{+) —} lim Ф ^{DN ( r ) —} Y n + d n ^, Ф ^DN (u n +1 ) ^— Y n +1 . r ^ u _n +                  2

(2.17)

In particular, Y n +1 > Y n + d n a/2 for all n — 0,... , m . Here we agree that Y o — 0 , Y m +1 — + ^ю, d o — 0 .

<1 By Corollary 1 the function S n — S n (r) is continuous. In view of (2.15) we see that ^ D^N (r) is a continuous function on (u _n ,u n +1 ]. If this function is not strictly monotone then we can find values r 1 , r 2 such that u _n < r 1 < r 2 < u n +1 and that Ф ^DN (r 1 ) — Ф ^DN (r 2 ). Then the functions u r ¹ (S), u r ² (S) are different self-similar solutions of type n to Stefan-Neumann

problem (1.1), (1.2), (2.1) with the same Neumann data b N = - Ф D^N (r i ) = - Ф D^N (r ₂ ), which contradicts to the uniqueness of this solution. Thus, Ф D^N (r) is a strictly monotone function.

If n < m, then

Ф D^N (u n +i ) =

k n F ‘(an) (u n+1 - U n )

^F ( *£ 1 ) - F ( a n ))

= ^Y n +1 -

In the case n = m

* DN (r)

k m F ‘ (am) ^{( r} — u m ) a m (F ( a m ) - F ( a m ))

k F ^{′ α}

> ----- a mj (r - u _m ) -^ . .

am          r^+м and we claim that

^ m^N (U m+1 ) =    lim    ^ m^N (r) = Y m +1 = + » .

r ^ U _m +1 = + m

We notice that by the strict convexity of function (2.10) its minimum point ξn depends continuously on the parameter s = -bN. In particular, the last coordinate ^n = ^n(s) is a continuous function of s, and the introduced in (2.7) Neumann-to-Dirichlet map ФND is a continuous function. As follows from (2.14), ^nYn + dna/2) = a while ^n(s) > a for s > Yn + dna/2. Therefore, for sufficiently small h > 0 the value dnα rh = ФND ^n + — + h) G (un,un+i]

and r h ^ u _n as h ^ 0. Since Ф D^N (r h ) = Y n +d n a/2 + h and the function Ф D^N (r) is monotone, we conclude that Ф D^N (u _n +) = Y n + d n a/2. In the case n = 0 the equality Ф D^N (u o +) = Y o = 0 directly follows from expression (2.16). Remind that Ф D^N (r) > Y n + d n a/2 for r > u _n . In particular,

^Y n +1 = Ф D^{N ( u} n +i ) >  ^Y n . 2“ •

Therefore, the function Ф D^N (r) strictly increases. >

It follows from Lemma 2 that the Dirichlet-to-Neumann map

^DN(r) = ^DN(r) if un < r C un+i, n = 0,... ,m is a strictly increasing function on (uo, +to) with the image

m

Ima = J (?n +    , Yn+1 ] • n=o

This function is continuous, except the points U i , i = 1,...,m, where it has the jumps Ф ^DN (u i +) - Ф ^DN (u i - ) = d i a/2. In particular, Im _a = (0, +ot) and Ф ^DN (r) is a homeomorphism of (u o , +ot) onto (0, +ot) if either a = 0 or all d i = 0. In general case, Theorem 2 implies the following results on correctness of problem (1.1), (1.2), (2.1).

Theorem 3. A solution of the Stefan–Neumann problem (1.1) , (1.2) , (2.1) is unique and exists if and only if - b N ∈ Im _α . The type n of this solution is determined by the condition Y n + d n ^{a/ 2} <  ^{- b} N ^C Y n +1 .

In the case m =1 the parameter y ₁ = ^{k 0} )u (-uF (F/a,/) a 0 ) • Hence, for 0 <  -b N C y ₁ a unique solution u = u(^) of (1.1), (1.2), (2.1) has type 0 while for -b N > Y i + d i a/2 it has type 1. In the remaining case Y i <  -b N C Y i + d i a/2 solutions are absent.

Remark 3. In the case a = 0 we obtain the standard Stefan-Neumann problem (1.1), (1.2), (2.1) in a half-line x >  0. As follows from Theorem 3, this problem has a unique solution (2.5), (2.6), where the type n is determined from the relation Y n <  -b N C Y n +1 -

3.    The Case of Infinitely Many Phase Transitions

In this section we consider the exotic case when the number m of phase transitions is infinite. More precisely, we suppose that the phase transitions temperatures U i >  u q form a strictly increasing sequence, u . +i > U i , i G N. In (1.1) there are infinitely many phases, the i-th phase corresponds to the temperature u G ( u . ,u .+i ) , i = { 0 } U N. The parameters a i ,k i >  0 are, respectively, the diffusivity constant and the thermal conductivity of the i-th phase, i = 0,1,...; d i ^ 0 is the latent specific heat of the i-th phase transition, i = 1, 2,... We will study the problem (1.1), (1.2), (1.4) in the quarter-plane П о with possibly infinite Dirichlet data u d = lim^ ^ U i C + ^ . A self-similar solution u = u(£ ), £ = x/y/t, of this problem is given by expression (1.5), where now i runs over all nonnegative integers. The Stefan conditions (1.2) reduce again to (now infinite) system (2.1), which can be written in the form (d/d£ i )E(£) = 0, where the functional

• E ) = — E k i    - ^ (F (I) - F (^)) + E d^ ,

  (3.1)

  
  

i=0                                                 i =1

I = (£ i ) i e N G Q = { £ = (£ i ) i e N G l ^ : £ >  &+1 > 0 ( V i G N) } .

Here, as usual, l ^ is the space of bounded sequences equipped with the norm ||£ ||_ю = sup |£ i | . As is well-known, this space is dual to the space of summable sequences l i . Observe also that there is only finite number of terms in (3.1) depending on a fixed variable ξ i . Therefore, the partial derivatives (d/d£ i )E(^) are well defined whenever the value E(£ ) is finite.

We do not include the natural requirement lim i ,^ £ = 0 in the definition of Q because this requirement spoils coercivity of the functional E in the weak- ∗ topology. Observe that the functional E may take the value + w , moreover, it can happen that E = + w . Assuming that the latter does not however happen, i. e. the functional E is proper, E = + ro , we will show that this functional admits a unique global minimum point. For that we need some nice properties of E collected below.

Proposition 4. (i) The functional E(£ ) is low semi-continuous in weak-* topology;

• (ii)    It is coercive, that is, the sets Q(c) = { £ G Q | E(£ ) C c} are weakly-* compact;

• (iii)    The functional E(£ ) is strictly convex on Q .

• <1 It is known that weak- * convergence £_ri ^^ of a sequence £_ri = (£ n ) i _e N G l ^ , n G N, is equivalent to uniform boundedness | £ n | C const and elementwise convergence Q ^ £ of ^{i                                      i} n →∞

this sequence. Assume that П_п G Q, £ G Q, and П_п ^ £ weakly- * in l ^ . Then £ П ^ £ as n → ∞ for each i ∈ N. Since all terms in formula (3.1) are nonnegative, we can apply Fatou’s lemma for sums and conclude that

^{E (} ^ = £ — k^+i — i ln (Fg) — F^))^{+ E} d& i =0                                                     i =1

^C l"' {£ ^{- k} i (“ - + ^{1 -} • ^ln ( F ( 0- ) ^- F ( ' )) ⁺ E dr¹- } =>“ E& t

Hence, the functional E is weakly- ∗ low semi-continuous, and (i) is proven.

To prove (ii) we first notice that Q(c) = 0 if c £ 0 and we can suppose that c > 0. If £ = (£ i ) i e N € Q(c) then relations (1.8) hold for all i G { 0 } U N. As in the proof of Proposition 1, we derive from these relations that

£ 1 <  r ₂ = a ₀ F ^-i ( 1 - e ^- * .'   . ) .

Further, it follows from (1.8) that for every i G N

F (~ ^ — F ( ~) ^ ^ i = ^eXp ( —     --- Г \

\ai)      a Ui /              \   ki(ui +i - Ui )/

Since F (£) is Lipschitz with constant 1, this implies the inequalities

£ i - £ i+i >  a i ^ i >  0, i G N.

Hence, Q(c) is contained in the set

K = { £ G l ^ : 0 <  £ i £ Г 2 ,£ i - £ i+i >  a i S i ( V i G N) } .

Obviously, this set is bounded and weakly- * closed in l ^ . By the Banach-Alaoglu theorem the set K is weakly- * compact. We notice that sequences £ G K strictly decrease, £ i > £ i ₊i ^ 0 for all i G N, and K C Q. Since E is weakly- * semi-continuous, the set Q(c) is a closed subset of K and, therefore, is compact. Coercivity of E is proved.

To prove (iii), we observe that the functional E(£) is convex as a sum of the convex functionals -k i (u i+i — U i )ln(F(£ i /a i ) — F (£ i +i /a i )) and d i £ 2 /4. By the same reason for each m G N the functional

R m (£ ) =

∞

- ^ k i (u i₊i i=m

- u i ) ln

(F^ -     ))+£

4 4 7        4      7 7     i=m+i

is convex while the function

E _m ξ _m

= - E ^+i - u i )ln (f(|) - f(^ ))₊ E <  i =0                            ^{7       4     77}    i =i

£ _m = ( £ i , . . . , £ _m )

In the case E(f) = + w Proposition 4 allows to establish existence of a solution to the problem (1.1), (1.2), (1.4) with infinite number of phase transitions. More precisely, the following statement holds.

Theorem 4. Assume that the potential E(f ) is a proper functional, i. e. N = inf E(f) <  +to . Then there exists a unique minimum point f ⁰ = (f ° ) i e N € Q of E(f ) , i.e., E(f ⁰ ) = N. Moreover, lim i ^_№ f ⁰ = 0 , and function (1.5)

• <1 We define for n € N the sub-level sets K _n = Q(N + 1/n), where E <  N + 1/n. Then these sets are nonempty. By Proposition 4 they are weakly- * compact. Obviously, K n +i C K _n for all n € N. By Cantor’s intersection theorem there exists a point f ⁰ € Hn e N K n . Then, N <  E(f ⁰ ) <  N + 1/n for all n € N, which implies that E(f ⁰ ) = N , and f ⁰ is a point of global minimum of E . Uniqueness of this point directly follows from the strict convexity of E stated in Proposition 4 (iii). It only remains to prove that the sequence f ⁰ = (f ° ) i e N vanishes. Assuming the contrary, we will have lim^ ^ f ⁰ = inf i _e N f ⁰ = r >  0. Then the sequence f ^r = f ⁰ — r with components f 0 — r lies in Q. Since 1 — F ((f 0 — r)/a 0 ) > 1 — F (f 0 )/° 0 and for all i € N (f ⁰ — r) ² < (f ⁰ ) ² ,

( l O - r ’ /a i $/ ^{a i}

F(^) — F(^) / e-4 ds> / e-4 ds = F(I) — F(')’ (€0+i—r’/a I0+i/ai all the terms in expression (3.1) became smaller if we replace f0 by fr. Therefore, E(fr) < E(f0) = N, which is impossible. Hence, limi ,^ f0 = 0. Since f0 is a minimum point of E then (d/dfi)E(f0) = 0 for all i € N and Stefan conditions (1.2) are satisfied. We conclude that (3.2) is a solution of our problem (1.1), (1.2), (1.4). >

Example. Assume that k i = a ² = 1, i ^ 0; d i = 0, i >  0. Then Stefan conditions (1.2) simply means that u(f) is a self-similar solution of the heat equation u _t = u _xx . Therefore, u(f) = CF (f) + C 2 , Ci, C 2 = const, and u(f) is a bounded function. In particular, a solution of (1.1), (1.2), (1.4) with u d = lim i ,^ U i = + w does not exist. As is easy to verify, in this case E(f ) = + w , so that the assumption of Theorem 4 is violated. On the other hand, if u d <  + w then a unique solution of (1.1), (1.2), (1.4) has the form u(f) = 2u d — U 0 — 2( u d — U 0 )F (f). Solving the equations u(f) = u i , we find f i = F ^{- 1} ((2u d — U 0 — U i )/(2u D — 2U 0 )). Hence,

∞∞

E(f) = — ^(u i +i — U i )ln(F (f i ) — F (f i +i )) =£(U i ₊i — U i )(ln(2(u D — U 0 )) — ln(u i ₊i — U i )) i =0                                       i =0

∞

= (u d — u 0 ) ln(2(u D — u 0 )) — £(u i +i — u i ) ln(u i +i — u i ).

i =0

As follows from Theorem 4 and the uniqueness of our solution, this value is the minimum value of E whenever the functional E(f) is proper. In particular, E(f) = + w if (and only if) the series ^^(u i +i — u i ) ln(u i +i — u i ) is divergent. Choosing u i in such a way that

∞

^(u i +i — u i ) = u d — u 0 <  + ro , i =0

∞

^(u i +i — u i ) ln(u i +i — u i ) = +to , i =0

we find that the condition E(^) = + w is not necessary for existence of a solution to the problem (1.1), (1.2), (1.4).

Concerning the uniqueness, the following result holds.

• Theorem 5. Suppose that

∞

^ k i (u i+i - u i ) <  + w .                            (3.3)

i=0

Then a solution (3.2) of problem (1.1) , (1.2) , (1.4) is unique.

• <1 Let u¹^), u ² (^) be two solutions (3.2) of problem (1.1), (1.2), (1.4). Suppose that u ¹ (^ * ) = u ² (^ * ) at some point £ * > 0. Changing the places u ¹ , u ² if necessary, we may assume that u ¹ (^ * ) > u ² (C * ). We notice that K(u r (£)), r = 1, 2, are absolutely continuous decreasing functions,

∞∞

K(u r )(0+) - K(u r )(+to)= K (u d -) - K(u o ) = ^ (K (u i+i ) - K(u i ))=^k i (u i+1 - u i ) < +ю. i=0                       i=0

Since

ξ ∗

I ( K ( u ¹ )'^ ) - K(u ² ) ‘ (£)) d. = K ( u ¹ ) (^ * ) - K(u ² )^) >  0, 0

K(u¹)'^) - K(u²)'^) >  0 on a set A C (0, ^ * ) of positive Lebesgue measure. Let a G A be a common Lebesgue point of the functions K(u ¹ )(£), K(u ² )'(£,) G L 1 ((0, + w )). Then the functions K(u r ), r = 1, 2, have classic derivatives K(u r )'(a) and K(u ¹ )'(a) > K(u²)'(a). By the construction the functions u ^r (^) are solutions of Stefan-Neumann problem (1.1), (1.2), (2.2) in the domain П _а with Neumann data b N = K(u^r)'(a) and with finite number of phase transitions. The latter follows from the fact that the number of u i that less than max(u ¹ (a),u ² (a)) <  u d is finite. By Lemma 2 the Dirichlet-to-Neumann map -K(u)'(a) ^ u(a) is strictly increasing. Since -K(u ¹ )'(a) < -K(u²)'(a), we find that u ¹ (a) < u ² (a). By Corollary 2 this implies that u¹^ * ) C u ² (^ _t ), which contradicts to our assumption. We conclude that u ¹ = u ² and a solution of problem (1.1), (1.2), (1.4) is unique. >

Remark that F(^ i /a i ) - F(£ i+1 /a i ) < F (+ ^ ) - F(0) = 1/2 and therefore - ln(F (^ i /a i ) - F(^ i+1 /a i )) >  ln2. This implies that

∞

In 2 ^ k i (u i+1 - u i ) <  inf E(l), i=0

and, in particular, condition (3.3) is always satisfied whenever E(^) is a proper functional.