343
where Q = la, b[ x ]0, S[ x ]0, T[ denotes a larger given space-time domain containing the
glacier. Note that the third condition in (1.19)v implies that the equation (1.18)v will be
verified in the ice region {u > O} which is not a priori given.
This formul~tion as a free boundary problem corresponds to the well-known obstacle
problem (see [R2], for instance) and requires additional boundary and initial conditions
u(a, s, t) = u(b, s, t) = 0 for (s, t) E ]0, S[ X ]0, T[ ,
u(x, 0, t) = Ul(X, t), IJ o.u(x, S, t) = 0 for (x, t) E la, b[ x 10, T[ ,
u(x, s, 0) = uo(x, s) for (x, s) E la, b[ X ]0, S[,
(1.20)
(1.21 )v
(1.22)
where Uo and Ul are nonnegative prescribed functions. We note that, in (1.21 )v, the "final"
condition at s = S is necessary only if IJ > 0 and it may be replaced by the Signorini
condition
u ~ 0, o.u ~ 0 and uo.u = 0 at s = S, (x,t) E]a,b[ x 10,T[,
which would mean that if we truncate the ice sheet at s = S (i.e. ul.=s > 0) then its slope
o.H will be equal to the base slope o.h at that end.
Considering a regular nonnegative test function v = v(x, s), (x, s) E n = la, b[ x ]0, S[,
from (1.19)v we obtain
1o(OtU +o.u - o;u -IJO;U - r)(v - u) ~ 0
and integrating by parts, taking (1.20) and (1.21) into account, this yields, for each t > 0,
1o(OtU + O.U - r) (v - u) + 10 o",uo",(v - u) + IJ 10 o.uo.(v - u) ~ 0 , (1.23)
for all v ~ 0 such that v = 0 at x = a, b and v = Ul at s = O.
In the form (1.23), known as a variational inequality, this mathematical problem is
well-posed, in what concerns the existence, uniqueness and continuous dependence of the
solution and the iced region {u > O}. Under appropriate assumptions, this will be shown
in the section 2, with particular emphasis to the asymptotic cases IJ -+ 0 and t -+ +00,
by exploiting and developing some mathematical analogies with the model of [RSI, in a
different physical context.
We observe that the variational inequality (1.23) condensates in fact four different
problems: two time dependent cases, the one corresponding to a positive parameter IJ > 0,
which is associated with a parabolic inequality, and the limit case IJ = 0, associated with a
ultraparabolic inequality (with two "times"); and two stationary (i.e., time independent)
problems, an elliptic one if IJ > 0 and a parabolic one if IJ = 0, where the rescaled
space variable s has the mathematical role of a "time" , both corresponding to asymptotic
states at "t = 00". As we shall show the convergences IJ -+ 0 and t -+ 00 are stable in
appropriate functional spaces for the respective solutions.
A different model for shallow-ice-sheets has been recently considered in [DFS] also as
an unilateral problem, but for a different nonlinear diffusion equation.
where Q = la, b[ x ]0, S[ x ]0, T[ denotes a larger given space-time domain containing the
glacier. Note that the third condition in (1.19)v implies that the equation (1.18)v will be
verified in the ice region {u > O} which is not a priori given.
This formul~tion as a free boundary problem corresponds to the well-known obstacle
problem (see [R2], for instance) and requires additional boundary and initial conditions
u(a, s, t) = u(b, s, t) = 0 for (s, t) E ]0, S[ X ]0, T[ ,
u(x, 0, t) = Ul(X, t), IJ o.u(x, S, t) = 0 for (x, t) E la, b[ x 10, T[ ,
u(x, s, 0) = uo(x, s) for (x, s) E la, b[ X ]0, S[,
(1.20)
(1.21 )v
(1.22)
where Uo and Ul are nonnegative prescribed functions. We note that, in (1.21 )v, the "final"
condition at s = S is necessary only if IJ > 0 and it may be replaced by the Signorini
condition
u ~ 0, o.u ~ 0 and uo.u = 0 at s = S, (x,t) E]a,b[ x 10,T[,
which would mean that if we truncate the ice sheet at s = S (i.e. ul.=s > 0) then its slope
o.H will be equal to the base slope o.h at that end.
Considering a regular nonnegative test function v = v(x, s), (x, s) E n = la, b[ x ]0, S[,
from (1.19)v we obtain
1o(OtU +o.u - o;u -IJO;U - r)(v - u) ~ 0
and integrating by parts, taking (1.20) and (1.21) into account, this yields, for each t > 0,
1o(OtU + O.U - r) (v - u) + 10 o",uo",(v - u) + IJ 10 o.uo.(v - u) ~ 0 , (1.23)
for all v ~ 0 such that v = 0 at x = a, b and v = Ul at s = O.
In the form (1.23), known as a variational inequality, this mathematical problem is
well-posed, in what concerns the existence, uniqueness and continuous dependence of the
solution and the iced region {u > O}. Under appropriate assumptions, this will be shown
in the section 2, with particular emphasis to the asymptotic cases IJ -+ 0 and t -+ +00,
by exploiting and developing some mathematical analogies with the model of [RSI, in a
different physical context.
We observe that the variational inequality (1.23) condensates in fact four different
problems: two time dependent cases, the one corresponding to a positive parameter IJ > 0,
which is associated with a parabolic inequality, and the limit case IJ = 0, associated with a
ultraparabolic inequality (with two "times"); and two stationary (i.e., time independent)
problems, an elliptic one if IJ > 0 and a parabolic one if IJ = 0, where the rescaled
space variable s has the mathematical role of a "time" , both corresponding to asymptotic
states at "t = 00". As we shall show the convergences IJ -+ 0 and t -+ 00 are stable in
appropriate functional spaces for the respective solutions.
A different model for shallow-ice-sheets has been recently considered in [DFS] also as
an unilateral problem, but for a different nonlinear diffusion equation.