230
On the other hand, from Proposition 3, (29) and (30) we know that v~ satisfies
1
Vt - (p(x)IVxI P - 2 vx)x + Bv::; -A + Q(J,(v) in I x (0, TA ),
p(x)lvxI P - 2 vx = °
on 81 x (0, TA ),
v(x,O) = ua(x)
on I.
and then by Theorem 2 we conclude that u' 2: VA on 1 x [0, TA]' Using the same kind of a
priori estimates as in Lemma 2 we have that u' --+ u (weakly in U(O, T : V) and weakly
in U"'(O, T : V)) as E ~ 0, with u a bounded weak solution of (P) such that
A
-
u 2: v on I x [0, TAl, for any A > 0,
(31 )
and the conclusion follows from (28).
•
Remark 3. It is not difficult to show (see Feireisl-Norbury [1991]) that (27) implies that
the solution u of Theorem 3 satisfies ux ( x, t) > ° for any x E (-1,0) U (0,1) and t > 0.
Then by the Implicit Function Theorem there exists a continuous function ( : [0, T] --+
[0, 1], defining completely the free boundary associated to u i.e. such that for any fixed
t E [O,T]
{x E 1: u(x, t) = I} = {-((tn U {((t)}.
(32)
Clearly ( E C1((0, T]). Moreover (31) implies that
((t) 2: tf A for any A > 0.
As ((0) = ° we deduce that necessarily ('(t) t +00 as t ~ 0.
3.3. On the uniqueness of solutions of the Budyko model.
We have proved that the mere presence of a "bad point" Xa where u(ta, xo ) = -10
and ux(ta, xa) = ° can be the reason of multiple solutions for t 2: ta. The following result
shows that if the initial datum Ua leads to a solution u never fiat at the level u = -10
then in fact u is the unique solution. We introduce the following notation:
Definition 2 . Let w E L=(I). We say that w satisfies the strong (resp. weak) pnondegeneracy property if there exists C > ° and Ea > ° such that for any E E (0, Ea)
I{x E I: Iw(x) + 101::; E}I ::; CE P - 1
(resp. I{x E I: ° < Iw(x) + 101::; E}I ::; CE P - 1 ).
Theorem 4 Assume p 2: 2. Let Re satisfying (10) and Ra given by (4) and (5). Let
Ua E L=(I).
(i) Assume that there exists a solution u(·, t) satisfying the strong p-nondegeneracy property for any t E [0, T]. Then u is the unique bounded weak solution of (P).
(ii) At most there is a unique solution among the class of bounded weak solutions satisfying
the weak p-nondegeneracy property.
On the other hand, from Proposition 3, (29) and (30) we know that v~ satisfies
1
Vt - (p(x)IVxI P - 2 vx)x + Bv::; -A + Q(J,(v) in I x (0, TA ),
p(x)lvxI P - 2 vx = °
on 81 x (0, TA ),
v(x,O) = ua(x)
on I.
and then by Theorem 2 we conclude that u' 2: VA on 1 x [0, TA]' Using the same kind of a
priori estimates as in Lemma 2 we have that u' --+ u (weakly in U(O, T : V) and weakly
in U"'(O, T : V)) as E ~ 0, with u a bounded weak solution of (P) such that
A
-
u 2: v on I x [0, TAl, for any A > 0,
(31 )
and the conclusion follows from (28).
•
Remark 3. It is not difficult to show (see Feireisl-Norbury [1991]) that (27) implies that
the solution u of Theorem 3 satisfies ux ( x, t) > ° for any x E (-1,0) U (0,1) and t > 0.
Then by the Implicit Function Theorem there exists a continuous function ( : [0, T] --+
[0, 1], defining completely the free boundary associated to u i.e. such that for any fixed
t E [O,T]
{x E 1: u(x, t) = I} = {-((tn U {((t)}.
(32)
Clearly ( E C1((0, T]). Moreover (31) implies that
((t) 2: tf A for any A > 0.
As ((0) = ° we deduce that necessarily ('(t) t +00 as t ~ 0.
3.3. On the uniqueness of solutions of the Budyko model.
We have proved that the mere presence of a "bad point" Xa where u(ta, xo ) = -10
and ux(ta, xa) = ° can be the reason of multiple solutions for t 2: ta. The following result
shows that if the initial datum Ua leads to a solution u never fiat at the level u = -10
then in fact u is the unique solution. We introduce the following notation:
Definition 2 . Let w E L=(I). We say that w satisfies the strong (resp. weak) pnondegeneracy property if there exists C > ° and Ea > ° such that for any E E (0, Ea)
I{x E I: Iw(x) + 101::; E}I ::; CE P - 1
(resp. I{x E I: ° < Iw(x) + 101::; E}I ::; CE P - 1 ).
Theorem 4 Assume p 2: 2. Let Re satisfying (10) and Ra given by (4) and (5). Let
Ua E L=(I).
(i) Assume that there exists a solution u(·, t) satisfying the strong p-nondegeneracy property for any t E [0, T]. Then u is the unique bounded weak solution of (P).
(ii) At most there is a unique solution among the class of bounded weak solutions satisfying
the weak p-nondegeneracy property.