357
Since X*(t) is uniformly bounded, X*(t) -- X, for a certain X, ° ~ X ~ 1, when
t -+ +00. So, /*(t) X*(t) = U*(t) - f;"') x*(t) + f;'" x*(t) converges to f;'" x~ and to
f;'" x· Since foo t= ° a.e., we have X = X~ a.e. and so X*(t) -" X~ when t -+ 00 in
£00(0,1; £OO(O))-weak*, first, and then also strongly in V(O, 1; V(O)), for p E [1,00[,
afterwards. I
3 - Analysis of the Shallow-Ice Temperature with Phase Change
In the first subsection we consider the parabolic two-phase Stefan problem, corresponding to the steady-state temperature in the shallow-ice model. This is done by recalling
the variational inequality formulation and well known results.
In the subsection 2 we consider the evolutionary two-phase Stefan limit (1/ = 0) problem. We obtain a ultraparabolic variational inequality formulation of this problem and
we prove the existence and the uniqueness of the solution.
In the last subsection we study the asymptotic behaviour in time, proving that the solution of the evolutionary ultraparabolic problem converges to the solution of the parabolic
stationary problem, when t -+ +00.
We follow here also the variational inequality approach which yields new results for
the ultraparabolic problem. However we observe that other approaches are also possible
as in the classical problem (see, for instance, [R3J, for references).
We fix some notations: (x, z) E w, (t, s) E R, where
w is an open, smooth subset of R2 , R = ]0, T[ x ]0, S[ ,
0= wx ]0, s[, Q = w x R ,
A = at + a. , b.' = a; + a; .
Note that 0 and Q are in general different from the previous section. However, in the
special case in which we can neglect the dependence in z and w = la, b[, they coincide.
All the results in this section still hold for that lower dimensional case, with obvious
simplifications.
3.1. The parabolic steady-state problem
Let H be the Heaviside graph, more precisely,
b a Lipschitz real function satisfying
if s < 0,
if s = 0,
if s > ° ,
b(O} = 0, 0< b. ~ b'(S} ~ b' a.e. in R ,
(3.1)
{3.2}
8~, hoo and a are given functions, being the first one defined on w, hoo on aw x [0, S] and
a on aw.
Since X*(t) is uniformly bounded, X*(t) -- X, for a certain X, ° ~ X ~ 1, when
t -+ +00. So, /*(t) X*(t) = U*(t) - f;"') x*(t) + f;'" x*(t) converges to f;'" x~ and to
f;'" x· Since foo t= ° a.e., we have X = X~ a.e. and so X*(t) -" X~ when t -+ 00 in
£00(0,1; £OO(O))-weak*, first, and then also strongly in V(O, 1; V(O)), for p E [1,00[,
afterwards. I
3 - Analysis of the Shallow-Ice Temperature with Phase Change
In the first subsection we consider the parabolic two-phase Stefan problem, corresponding to the steady-state temperature in the shallow-ice model. This is done by recalling
the variational inequality formulation and well known results.
In the subsection 2 we consider the evolutionary two-phase Stefan limit (1/ = 0) problem. We obtain a ultraparabolic variational inequality formulation of this problem and
we prove the existence and the uniqueness of the solution.
In the last subsection we study the asymptotic behaviour in time, proving that the solution of the evolutionary ultraparabolic problem converges to the solution of the parabolic
stationary problem, when t -+ +00.
We follow here also the variational inequality approach which yields new results for
the ultraparabolic problem. However we observe that other approaches are also possible
as in the classical problem (see, for instance, [R3J, for references).
We fix some notations: (x, z) E w, (t, s) E R, where
w is an open, smooth subset of R2 , R = ]0, T[ x ]0, S[ ,
0= wx ]0, s[, Q = w x R ,
A = at + a. , b.' = a; + a; .
Note that 0 and Q are in general different from the previous section. However, in the
special case in which we can neglect the dependence in z and w = la, b[, they coincide.
All the results in this section still hold for that lower dimensional case, with obvious
simplifications.
3.1. The parabolic steady-state problem
Let H be the Heaviside graph, more precisely,
b a Lipschitz real function satisfying
if s < 0,
if s = 0,
if s > ° ,
b(O} = 0, 0< b. ~ b'(S} ~ b' a.e. in R ,
(3.1)
{3.2}
8~, hoo and a are given functions, being the first one defined on w, hoo on aw x [0, S] and
a on aw.