227
Proof. It suffices to use now (w - w*)+ (= max(w - w*,O)) as a test function. Indeed,
by a variant of a result due to Stampacchia we know that (w - w*)+ E LP(O, T : V).
Moreover
< Wt(t) - w;(t), (w(t) - w*(t))+ >v'v= ~ hl[w(t) - w*(t)J+1 2 dx
and inequality (25) follows.
•
3.2. A non uniqueness result for the Budyko model.
The discontinuity of the coalbedo function ;3( u) and its role as a source term in the
equation may lead to the existence of multiple (even infinite) solutions of the problem.
This has already been shown in Dfaz [1992] for the case of the homogeneous (zerodimensional) balance model
dtt
dt = Ra(tt) - Re(tt).
The main purpose of this subsection is to show that this situation may also occurs for
problem (P). Our pre8entation is inspired in the work of Feireisl-Norbury [1991] (see also
Feireisl [1991]). We fix our attention in the special case of Budyko model i.e., Ra and Re
are given by (4), (5) and (7) respectively. We shall also assume that
Q(x, t) == Q and Qai < A - lOB.
Consider a function tto such that
tto E eoo(I), tto(x) = tto( -x) for all x E [0,1], }
tt~k)(O) = ° for k = 1,2, tto(O) = -10
tt~(x) < ° if x E (0, 1), tt~(I) = °
(26)
(27)
(in this hypothetical case the maximum of the distributed temperature is -lO o e and it is
only attained at the equator). We first show the existence of a "completely ice covered"
solution tt*.
Proposition 2 . Let Ra, Re given by (4), (5) and (7) respectively. Assume that (26) and
(27) holds. Then there exist at least one solution u* of (P) such that tt*(x, t) < -10 for
any x E (-1,1) andt E (O,T].
Proof. Let tt* be the unique solution of the problem
1
ttl - (p(x)lttx I P - 2 ttx)x + Btt = -A + Qai'
(P) p( x )lttx IP-2ttx = °
tt(x, 0) = tto(x)
x E I, t > 0,
x E aI, t > 0,
x E I.
The existence and uniqueness can be shown again by different methods (for instance, it
is a trivial consequence of Proposition 1). The function z = -10 - tt* satisfies that
Proof. It suffices to use now (w - w*)+ (= max(w - w*,O)) as a test function. Indeed,
by a variant of a result due to Stampacchia we know that (w - w*)+ E LP(O, T : V).
Moreover
< Wt(t) - w;(t), (w(t) - w*(t))+ >v'v= ~ hl[w(t) - w*(t)J+1 2 dx
and inequality (25) follows.
•
3.2. A non uniqueness result for the Budyko model.
The discontinuity of the coalbedo function ;3( u) and its role as a source term in the
equation may lead to the existence of multiple (even infinite) solutions of the problem.
This has already been shown in Dfaz [1992] for the case of the homogeneous (zerodimensional) balance model
dtt
dt = Ra(tt) - Re(tt).
The main purpose of this subsection is to show that this situation may also occurs for
problem (P). Our pre8entation is inspired in the work of Feireisl-Norbury [1991] (see also
Feireisl [1991]). We fix our attention in the special case of Budyko model i.e., Ra and Re
are given by (4), (5) and (7) respectively. We shall also assume that
Q(x, t) == Q and Qai < A - lOB.
Consider a function tto such that
tto E eoo(I), tto(x) = tto( -x) for all x E [0,1], }
tt~k)(O) = ° for k = 1,2, tto(O) = -10
tt~(x) < ° if x E (0, 1), tt~(I) = °
(26)
(27)
(in this hypothetical case the maximum of the distributed temperature is -lO o e and it is
only attained at the equator). We first show the existence of a "completely ice covered"
solution tt*.
Proposition 2 . Let Ra, Re given by (4), (5) and (7) respectively. Assume that (26) and
(27) holds. Then there exist at least one solution u* of (P) such that tt*(x, t) < -10 for
any x E (-1,1) andt E (O,T].
Proof. Let tt* be the unique solution of the problem
1
ttl - (p(x)lttx I P - 2 ttx)x + Btt = -A + Qai'
(P) p( x )lttx IP-2ttx = °
tt(x, 0) = tto(x)
x E I, t > 0,
x E aI, t > 0,
x E I.
The existence and uniqueness can be shown again by different methods (for instance, it
is a trivial consequence of Proposition 1). The function z = -10 - tt* satisfies that