225
Given t, m, nand k positive numbers we consider the problem (P,)
1
Ut - [p,(x)IU x I P - 2 ux ]x - tu xx = Qn(X, t)f3,(u) - Re(x, t, u), x E I x (0, T),
p,(x)(lux I P - 2 ux + tux) = °
on 01 x (0, T),
u(x,O) = UO,m(x)
on I.
The partial differential equation is now uniformly parabolic and so by well-known results (see e.g. Ladyzenskaja-Solonnikov-Ural'ceva [1968], Chapt.V) there exists a unique
classical solution U = u"m,n,k. In order to study the convergence, when € "" ° and
m, n, k -+ +00 we need some a priori estimates.
Lemma 2 . The solution U of (P,) satisfies (for nand k large enough)
II U liLoc(Ix(o,T))::; C,
II p,Ux IILP(O,T:LP(I))::; C,
where C denotes a positive constant independent of €, m, nand k.
(21)
(22)
Proof. Estimate (21) is derived from the maximum principle (see e.g. LadyzenskajaSolonnikov-Ural'ceva [1968]). To obtain (22) we multiply the equation by U. Integrating
by parts we obtain
~ ~ 1 U 2 (x, t)dx + 1 p,lUxl Pdx + € lIUxl 2dx ::; C
(where we have used (19) and (21)).
•
U sing the a priori estimates and the assumption (11) the proof of the convergence
U -+ u, f3,(U) -+ z with z E f3(u) and that u is a bounded weak solution of (P) is
standard (notice that this is not the case if we want obtain more regularity on Ut as, for
instance, that given in Corollary 1).
Remark 2. The regularization of the multivalued term f3(u) was already carryed out in
Xu [1991] for p = 2 (see also Feireisl-Norbury [1991] for some related problems). We also
point out that the existence of a weak solution can be obtained by the method of upper
and lower solutions combined with monotone iteration arguments (see e.g. Carl [1989]
and Dfaz-Stakgold [1989] for other related problems).
3. On the uniqueness of solutions: positive and negative answers.
The type of answer to the question of the uniqueness of solutions to problem (P) is
rather different in the cases of the Sellers model (where Ra(x, t, u) is a smooth function)
and the Budyko model (where Ra (x, t, u) is a discontinuous function of u).
Given t, m, nand k positive numbers we consider the problem (P,)
1
Ut - [p,(x)IU x I P - 2 ux ]x - tu xx = Qn(X, t)f3,(u) - Re(x, t, u), x E I x (0, T),
p,(x)(lux I P - 2 ux + tux) = °
on 01 x (0, T),
u(x,O) = UO,m(x)
on I.
The partial differential equation is now uniformly parabolic and so by well-known results (see e.g. Ladyzenskaja-Solonnikov-Ural'ceva [1968], Chapt.V) there exists a unique
classical solution U = u"m,n,k. In order to study the convergence, when € "" ° and
m, n, k -+ +00 we need some a priori estimates.
Lemma 2 . The solution U of (P,) satisfies (for nand k large enough)
II U liLoc(Ix(o,T))::; C,
II p,Ux IILP(O,T:LP(I))::; C,
where C denotes a positive constant independent of €, m, nand k.
(21)
(22)
Proof. Estimate (21) is derived from the maximum principle (see e.g. LadyzenskajaSolonnikov-Ural'ceva [1968]). To obtain (22) we multiply the equation by U. Integrating
by parts we obtain
~ ~ 1 U 2 (x, t)dx + 1 p,lUxl Pdx + € lIUxl 2dx ::; C
(where we have used (19) and (21)).
•
U sing the a priori estimates and the assumption (11) the proof of the convergence
U -+ u, f3,(U) -+ z with z E f3(u) and that u is a bounded weak solution of (P) is
standard (notice that this is not the case if we want obtain more regularity on Ut as, for
instance, that given in Corollary 1).
Remark 2. The regularization of the multivalued term f3(u) was already carryed out in
Xu [1991] for p = 2 (see also Feireisl-Norbury [1991] for some related problems). We also
point out that the existence of a weak solution can be obtained by the method of upper
and lower solutions combined with monotone iteration arguments (see e.g. Carl [1989]
and Dfaz-Stakgold [1989] for other related problems).
3. On the uniqueness of solutions: positive and negative answers.
The type of answer to the question of the uniqueness of solutions to problem (P) is
rather different in the cases of the Sellers model (where Ra(x, t, u) is a smooth function)
and the Budyko model (where Ra (x, t, u) is a discontinuous function of u).