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Corollary 1 . Assume (8), (10), (II), (14) and p ~ 2. Then for any uo E L2(1) there
exists a function u E C([O,T] : L2(/)) such that u(t) E D(A) a.e. t E (O,T], d~ E
L 2 (0,T: L2(/)), rp(u) E L 1 (0,T : JR) and it satisfies (P) a.e. t E (O,T) on L2(1)
as well as in the sense of (12). Moreover, if Uo E V then ~ E L2(0, T : L2(/)) and
u E C([O, T] : V). Finally, if Uo E Loo(1) then u E Loo(/ x (0, T)).
Proof. The existence of u satisfying (P) a.e. t E (0, T) on L2(1) is a consequence of the
application of a suitable fixed theorem for a compact operator (see, e.g., Vrabie [1987],
Corollary 2.3.2). The application of such results is guaranteed by Proposition 1, Lemma
1 and the assumptions (10) and (11). This function obviously satisfies trivially (12) (take
integrals on (7, T) x I and make 7 \t 0). The boundedness of u, assumed Uo E Loo(/), is
proved as in Proposition 1 if the right hand side of the equation is a bounded term .
•
Remark 1. The above method can be applied to two-dimensional problems (on a compact
Riemannian manifold without boundary): see Hetzer [1990] (for the Sellers type model)
and Dfaz-Tello [1993], [1996] when c == 1 and Bermejo - Diaz - Tello [1996] when c E
Loo(M) (for the Budyko model).
2.2. Existence via a regularization method.
The existence of a bounded weak solution of (P) can be also obtained by approximating
the multivalued (discontinuous) term flO by a regular function fl, E coo(1R) with the
properties
fl:(s) ~ ° and Ifl,(s) I ~ M Vs E JR.
(19)
It is also usefull to remove the degeneracy at a I by replacing p( x) by
p,(x) = p(x) + t.
(20)
In order to approximate u by classical solutions of a related problem we also replace the
data Uo, Q and Re by Coo functions UO,m, Qn, Re,k such that
and
UO,m ----t Uo in L2(1), as m -+ 00,
Qn ----t Q in C(1 x [0, Tn,
Re,k satisfies (4), Re,k(·, ·,u) ----t Re(·, ·,u) in C(1 x [0, Tn }
for any fixed U E JR and Re,k(X, t,·) ----t Re(x, t,·) in C(J) for
any compact J c JR and any fixed (x, t) E 1 x [0, T].
Corollary 1 . Assume (8), (10), (II), (14) and p ~ 2. Then for any uo E L2(1) there
exists a function u E C([O,T] : L2(/)) such that u(t) E D(A) a.e. t E (O,T], d~ E
L 2 (0,T: L2(/)), rp(u) E L 1 (0,T : JR) and it satisfies (P) a.e. t E (O,T) on L2(1)
as well as in the sense of (12). Moreover, if Uo E V then ~ E L2(0, T : L2(/)) and
u E C([O, T] : V). Finally, if Uo E Loo(1) then u E Loo(/ x (0, T)).
Proof. The existence of u satisfying (P) a.e. t E (0, T) on L2(1) is a consequence of the
application of a suitable fixed theorem for a compact operator (see, e.g., Vrabie [1987],
Corollary 2.3.2). The application of such results is guaranteed by Proposition 1, Lemma
1 and the assumptions (10) and (11). This function obviously satisfies trivially (12) (take
integrals on (7, T) x I and make 7 \t 0). The boundedness of u, assumed Uo E Loo(/), is
proved as in Proposition 1 if the right hand side of the equation is a bounded term .
•
Remark 1. The above method can be applied to two-dimensional problems (on a compact
Riemannian manifold without boundary): see Hetzer [1990] (for the Sellers type model)
and Dfaz-Tello [1993], [1996] when c == 1 and Bermejo - Diaz - Tello [1996] when c E
Loo(M) (for the Budyko model).
2.2. Existence via a regularization method.
The existence of a bounded weak solution of (P) can be also obtained by approximating
the multivalued (discontinuous) term flO by a regular function fl, E coo(1R) with the
properties
fl:(s) ~ ° and Ifl,(s) I ~ M Vs E JR.
(19)
It is also usefull to remove the degeneracy at a I by replacing p( x) by
p,(x) = p(x) + t.
(20)
In order to approximate u by classical solutions of a related problem we also replace the
data Uo, Q and Re by Coo functions UO,m, Qn, Re,k such that
and
UO,m ----t Uo in L2(1), as m -+ 00,
Qn ----t Q in C(1 x [0, Tn,
Re,k satisfies (4), Re,k(·, ·,u) ----t Re(·, ·,u) in C(1 x [0, Tn }
for any fixed U E JR and Re,k(X, t,·) ----t Re(x, t,·) in C(J) for
any compact J c JR and any fixed (x, t) E 1 x [0, T].