345
We must add the boundary conditions, that may be of the form
ao
- = o:(h - 0) on aw X R ,
an
i.e., of Neumann type, where 0: = o:(x, z) > ° is a thermal coefficient, possibly discontinuous, and h = h(x, z, s, t) represents an external temperature, for instance of the air
or of the earth in contact with the glacier. Here a/an = \7' . n is the outward normal
derivative associated with the gradient \7' = (ax, az )'
Finally, we need to prescribe the Cauchy conditions, for given functions 0 0 and 0 1 , i.e.
In fact, also 1711=0 and 1718=0 must be known, but due to the relation (1.25), it is often
possible to take
1711=0 = b(OO) + ,\ X{8o>0} and 1718=0 = b(OI) + ,\ X{8I>0} ,
what we shall do here in the Section 3, where this problem is solved in a weak sense with
the variational inequalities method.
Actually we have an evolutionary problem, corresponding to the ultraparabolic equation (1.26) and a time independent problem, that corresponds to its asymptotic limit
"t = 00" and may be regarded as the steady-state solution of (1.26).
2 - Mathematical Analysis of the Glacier Kinematics
In the first subsection we treat the steady-state case as an elliptic obstacle problem. We
prove the existence of a regular solution u~ of the steady-state problem and we study the
behavior of u~ and of X{u:;.,>O}, when lJ -t 0, obtaining in the limit a parabolic variational
inequality.
In the subsection 2 we consider the evolutionary problem, studying also the behavior
of its solution u" and of X{u">O}, when lJ - t 0, remarking that the limit problem is
ultraparabolic.
In the third subsection, we study the asymptotic behaviour, when t --+ +00.
Here we extend the approach of [RS], based in a continuous casting model studied in
[RIJ, which was slightly simpler due to the homogeneous Cauchy data. It is interesting to
notice that, besides that one-phase Stefan problem, ultraparabolic problems also appear
in an entirely different framework in mathematical biology (see, for instance, [GLJ, [GLan]
and for a different approach also [K]).
2.1. The steady-state case
Let x Ela,b[, s ElO,S[ and 11 =la,b[xlO,S[, ro {a}xlO,s[U{b}xlO,s[, rl =
la, b[ x {a} and set
v = {v E HI(I1): vlro = o},
(2.1)
Koo = {v E v: v ?: 0, vir, = u~} ,
(2.2)
We must add the boundary conditions, that may be of the form
ao
- = o:(h - 0) on aw X R ,
an
i.e., of Neumann type, where 0: = o:(x, z) > ° is a thermal coefficient, possibly discontinuous, and h = h(x, z, s, t) represents an external temperature, for instance of the air
or of the earth in contact with the glacier. Here a/an = \7' . n is the outward normal
derivative associated with the gradient \7' = (ax, az )'
Finally, we need to prescribe the Cauchy conditions, for given functions 0 0 and 0 1 , i.e.
In fact, also 1711=0 and 1718=0 must be known, but due to the relation (1.25), it is often
possible to take
1711=0 = b(OO) + ,\ X{8o>0} and 1718=0 = b(OI) + ,\ X{8I>0} ,
what we shall do here in the Section 3, where this problem is solved in a weak sense with
the variational inequalities method.
Actually we have an evolutionary problem, corresponding to the ultraparabolic equation (1.26) and a time independent problem, that corresponds to its asymptotic limit
"t = 00" and may be regarded as the steady-state solution of (1.26).
2 - Mathematical Analysis of the Glacier Kinematics
In the first subsection we treat the steady-state case as an elliptic obstacle problem. We
prove the existence of a regular solution u~ of the steady-state problem and we study the
behavior of u~ and of X{u:;.,>O}, when lJ -t 0, obtaining in the limit a parabolic variational
inequality.
In the subsection 2 we consider the evolutionary problem, studying also the behavior
of its solution u" and of X{u">O}, when lJ - t 0, remarking that the limit problem is
ultraparabolic.
In the third subsection, we study the asymptotic behaviour, when t --+ +00.
Here we extend the approach of [RS], based in a continuous casting model studied in
[RIJ, which was slightly simpler due to the homogeneous Cauchy data. It is interesting to
notice that, besides that one-phase Stefan problem, ultraparabolic problems also appear
in an entirely different framework in mathematical biology (see, for instance, [GLJ, [GLan]
and for a different approach also [K]).
2.1. The steady-state case
Let x Ela,b[, s ElO,S[ and 11 =la,b[xlO,S[, ro {a}xlO,s[U{b}xlO,s[, rl =
la, b[ x {a} and set
v = {v E HI(I1): vlro = o},
(2.1)
Koo = {v E v: v ?: 0, vir, = u~} ,
(2.2)