364
with the same technique, by remarking that, for t > S ;::: s > 0 we have
f' r 0: [g(r, r + (t - s)) - goo(r)] 8r w(r, r + (t - s)) dr =
Jo Jaw
= r o:[g(s,t)-goo(s)] w(s,t)- f'r o:w(r,r+(t-s))8r [g(r,r-(t-s))-goo(r)] dr. 0
Jaw
Jo Jaw
REFERENCES
[DFS) DIAZ, J.I., FOWLER, A.C. and SCHIAVI, E. - Mathematical Analysis of a Shallow Ice·Sheet
Flow Model (to appear).
[F1) FOWLER, A.C. - Modelling ice-sheet dynamics, Geophys. Astrophys. Fluid Dynamics, 63
(1992), 29-65.
[F2) FOWLER, A.C. - Glacier and Ice Sheets, in this volume.
[GL) GARRONI, M.G. and LAMBERTI, L. - A variational problem for population dynamics with a
unilateral constraint, Boll. U.M.I., 16B (1979),876-896.
[GLan) GARRONI, M.G. and LANGLAIS, M. - Age dependent population diffusion with external contraint, J. Math. Bioi., 14 (1982), 77-94.
[HH) VAN HARTEN, A. and VAN HASSEL, R.R. - On a singularly perturbed time-dependent free
boundary problem, J. Math. Anal. and Appl., 137 (1989), 70-98.
[Hi) HINDMARSH, R.C.A. - Time-scales and degrees of freedom operating in the evolution of continental ice-sheets, Transl. Royal Soc. Edinburgh: Earth Sci., 81 (1990),371-384.
[Hu] HUTTER, K. - Theoretical Glaciology, D. Reidel, Dordrecht, 1983.
[Huy] HUYBRECHTS, P. - The Antartic ice-sheet and environmental change: a three-dimensional
modelling study, Ber. Polarforsch #99, Bremerhaven, 1992.
[K] KUBo, M. - Subdifferential operator approach to nonlinear age-dependent population dynamics, Japan J. Appl. Math., 5 (1988), 225-256.
[LSU) LADYZENSKAYA, O.A., SOLONNIKOV, V.A. and URAL'CEVA, N.N. - Linear and quasilinear
equations of parabolic type, Amer. Math. Soc. Trans!. Monog., 23, Providence, 1968.
[L1] LIONS, J.L. - Sur Ie Controle Optimale de Systemes Governes par des Equations aux Derivees
Partielles, Dunod, Paris, 1968.
[L2) LIONS, J .L. - Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires,
Dunod, Paris, 1969.
[L3] LIONS, J .L. - Sur Quelques Questions d'Analyse, de Mechanique et de Controle Optimal,
Presses Univ. Montreal, 1976.
[M) MORLAND, L.W. - The flow of ice sheets and ice shelves, IUTAM Internat. Summer School on
Continuum Mech. in Environmental Sci. and GeoPhys., Udine, June 1992.
[PO] PEIXOTO, J.P. and OORT, A.H. - Physics of Climate, American Inst. of Physics, N.Y., 1992.
[R1) RODRIGUES, J .F. - An evolutionary continuous casting problem of Stefan type, Quart. Appl.
Math., XLIV(1) (1986) 109-131.
[R2] RODRIGUES, J.F. - Obstacle Problems in Mathematical Physics, North Holland, Amsterdam,
1987.
[R3] RODRIGUES, J.F. - Variational methods in the Stefan problem, Lecture Notes in Mathematics,
#1584, Springer, Berlin, 1994, 147-212.
[RS) RODRIGUES, J.F. and SANTOS, L. - Asymptotic convergences in a one-phase continuous casting Stefan problem with high extraction velocity, IMA Journal of Appl. Math., 42 (1989),
251-267.
[V] VERBITSKY, M.Y. - Equilibrium ice-sheet scaling in climate modeling, Climate Dynamics, 7
(1992), 105-110.
[WL) WEHANSEN, J.V. and LAITONE, E.V. - Surface Waves, "Handbuch der Physik" (S. Fliigge,
ed.), Vo!' IX (Fluid Dynamics III), 1960,446-758.
with the same technique, by remarking that, for t > S ;::: s > 0 we have
f' r 0: [g(r, r + (t - s)) - goo(r)] 8r w(r, r + (t - s)) dr =
Jo Jaw
= r o:[g(s,t)-goo(s)] w(s,t)- f'r o:w(r,r+(t-s))8r [g(r,r-(t-s))-goo(r)] dr. 0
Jaw
Jo Jaw
REFERENCES
[DFS) DIAZ, J.I., FOWLER, A.C. and SCHIAVI, E. - Mathematical Analysis of a Shallow Ice·Sheet
Flow Model (to appear).
[F1) FOWLER, A.C. - Modelling ice-sheet dynamics, Geophys. Astrophys. Fluid Dynamics, 63
(1992), 29-65.
[F2) FOWLER, A.C. - Glacier and Ice Sheets, in this volume.
[GL) GARRONI, M.G. and LAMBERTI, L. - A variational problem for population dynamics with a
unilateral constraint, Boll. U.M.I., 16B (1979),876-896.
[GLan) GARRONI, M.G. and LANGLAIS, M. - Age dependent population diffusion with external contraint, J. Math. Bioi., 14 (1982), 77-94.
[HH) VAN HARTEN, A. and VAN HASSEL, R.R. - On a singularly perturbed time-dependent free
boundary problem, J. Math. Anal. and Appl., 137 (1989), 70-98.
[Hi) HINDMARSH, R.C.A. - Time-scales and degrees of freedom operating in the evolution of continental ice-sheets, Transl. Royal Soc. Edinburgh: Earth Sci., 81 (1990),371-384.
[Hu] HUTTER, K. - Theoretical Glaciology, D. Reidel, Dordrecht, 1983.
[Huy] HUYBRECHTS, P. - The Antartic ice-sheet and environmental change: a three-dimensional
modelling study, Ber. Polarforsch #99, Bremerhaven, 1992.
[K] KUBo, M. - Subdifferential operator approach to nonlinear age-dependent population dynamics, Japan J. Appl. Math., 5 (1988), 225-256.
[LSU) LADYZENSKAYA, O.A., SOLONNIKOV, V.A. and URAL'CEVA, N.N. - Linear and quasilinear
equations of parabolic type, Amer. Math. Soc. Trans!. Monog., 23, Providence, 1968.
[L1] LIONS, J.L. - Sur Ie Controle Optimale de Systemes Governes par des Equations aux Derivees
Partielles, Dunod, Paris, 1968.
[L2) LIONS, J .L. - Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires,
Dunod, Paris, 1969.
[L3] LIONS, J .L. - Sur Quelques Questions d'Analyse, de Mechanique et de Controle Optimal,
Presses Univ. Montreal, 1976.
[M) MORLAND, L.W. - The flow of ice sheets and ice shelves, IUTAM Internat. Summer School on
Continuum Mech. in Environmental Sci. and GeoPhys., Udine, June 1992.
[PO] PEIXOTO, J.P. and OORT, A.H. - Physics of Climate, American Inst. of Physics, N.Y., 1992.
[R1) RODRIGUES, J .F. - An evolutionary continuous casting problem of Stefan type, Quart. Appl.
Math., XLIV(1) (1986) 109-131.
[R2] RODRIGUES, J.F. - Obstacle Problems in Mathematical Physics, North Holland, Amsterdam,
1987.
[R3] RODRIGUES, J.F. - Variational methods in the Stefan problem, Lecture Notes in Mathematics,
#1584, Springer, Berlin, 1994, 147-212.
[RS) RODRIGUES, J.F. and SANTOS, L. - Asymptotic convergences in a one-phase continuous casting Stefan problem with high extraction velocity, IMA Journal of Appl. Math., 42 (1989),
251-267.
[V] VERBITSKY, M.Y. - Equilibrium ice-sheet scaling in climate modeling, Climate Dynamics, 7
(1992), 105-110.
[WL) WEHANSEN, J.V. and LAITONE, E.V. - Surface Waves, "Handbuch der Physik" (S. Fliigge,
ed.), Vo!' IX (Fluid Dynamics III), 1960,446-758.