272
the complexification of (D2I1)(/£, tI), Itt there exists a A E C with ( = e->' such that A
is an eigenvalue of
{
c(x )Otw( t, x) - div (k grad w( t,' ))( x) + [/£Q(t, x)( 020' )(x, u( t, x; Ii, 19))+
(14)
+g'(u(t,x;/£,19))]w(t,x) = AC(X)W(t,x)
x EM, t E (0,1)
w(l,·) = w(O, .).
Therefore (14) possesses a so-called principal eigenvalue AO(/£, 19) in the parabolic sense,
i.e. Ao(/£, 19) is real, Re >.. 2' : Ao(/£,19) for each eigenvalue A of (14), and ?R>" = Ao(/£,19)
implies A = >"0(/£,19) + i27r1 for some I E Z. The S-shapedness of I-l J relies as in Section
4. on the following two structural assumptions:
Structural Hypotheses.
(H'4) ?RA > 0 for all eigenvalues>.. of (14) with ?R>" > >"0(/£,19) and all (/£,19) E 1-lJ;
(H'5) W/£,19) E I-l J : AO(/£, 19) = O} < 00.
Now, we obtain the following version of the "S-shapedness Theorem", which has been
established in [Hetzer (1994)]:
Theorem 2. Let (HO), (H'l), (H2) and (H'3) - (H'5) be satisfied. Then
a) I-l J is the trace of a Jordan curve in R+ x C+(M): there exists a homeomorphism
, : R+ --+ I-l J (onto), which is C I from (0,00) onto I-l J \ {(O, On with ,'(p) i= (0,0) for
all p E (0,00).
b) I-l J is "S-shaped": prl o,(p) --+ 00, in! pr2 0 ,(p) --+ 00 as p --+ 00, and prl 0, has
an even number of strict local extrema.
c) Given (/£,19) E 1-lJ, u(·,·; Ii, 19) is an stable (unstable) l-periodic solution of (7) if
,'(,-1(/£,19)) > 0 « 0).
Thus, the essential consequence from a climatological point of view is that including
the seasonal cycle alone does not make a qualitative difference, though locally the forcing
(solar radiation flux) differs considerably from that of yearly averaged models.
The course of reasoning follows closely that outlined in Section 4. However, some
care is required in dealing with the a priori bound from below.
8. An A Priori Bound from Below and Uniqueness.
We need the following observation:
the complexification of (D2I1)(/£, tI), Itt there exists a A E C with ( = e->' such that A
is an eigenvalue of
{
c(x )Otw( t, x) - div (k grad w( t,' ))( x) + [/£Q(t, x)( 020' )(x, u( t, x; Ii, 19))+
(14)
+g'(u(t,x;/£,19))]w(t,x) = AC(X)W(t,x)
x EM, t E (0,1)
w(l,·) = w(O, .).
Therefore (14) possesses a so-called principal eigenvalue AO(/£, 19) in the parabolic sense,
i.e. Ao(/£, 19) is real, Re >.. 2' : Ao(/£,19) for each eigenvalue A of (14), and ?R>" = Ao(/£,19)
implies A = >"0(/£,19) + i27r1 for some I E Z. The S-shapedness of I-l J relies as in Section
4. on the following two structural assumptions:
Structural Hypotheses.
(H'4) ?RA > 0 for all eigenvalues>.. of (14) with ?R>" > >"0(/£,19) and all (/£,19) E 1-lJ;
(H'5) W/£,19) E I-l J : AO(/£, 19) = O} < 00.
Now, we obtain the following version of the "S-shapedness Theorem", which has been
established in [Hetzer (1994)]:
Theorem 2. Let (HO), (H'l), (H2) and (H'3) - (H'5) be satisfied. Then
a) I-l J is the trace of a Jordan curve in R+ x C+(M): there exists a homeomorphism
, : R+ --+ I-l J (onto), which is C I from (0,00) onto I-l J \ {(O, On with ,'(p) i= (0,0) for
all p E (0,00).
b) I-l J is "S-shaped": prl o,(p) --+ 00, in! pr2 0 ,(p) --+ 00 as p --+ 00, and prl 0, has
an even number of strict local extrema.
c) Given (/£,19) E 1-lJ, u(·,·; Ii, 19) is an stable (unstable) l-periodic solution of (7) if
,'(,-1(/£,19)) > 0 « 0).
Thus, the essential consequence from a climatological point of view is that including
the seasonal cycle alone does not make a qualitative difference, though locally the forcing
(solar radiation flux) differs considerably from that of yearly averaged models.
The course of reasoning follows closely that outlined in Section 4. However, some
care is required in dealing with the a priori bound from below.
8. An A Priori Bound from Below and Uniqueness.
We need the following observation: