269
for t E R+. Thus, b> 0 yields (01IT)(Jl,Y) = (028)(1; Jl,Y) > 0, hence for each Y E R+
there exists exactly one p,(y) with IT(jt(y),y) = y. The implicit function theorem shows
jt E C 2 ((0, 00), (0, 00)) and (10) implies jt to be continuous at y = 0 with value o.
Again, turning points of ~ are those (p,(y), y), where jt(y) a strict local extrema of
jt. We obtain by implicit differentation that
"() 1-(~IT)(jt(y),y)
Jl y = (01 IT)(p,(y), y)
Y E (0,00),
hence turning points of ~ occur at those zeroes of y f-+ 1 - (02II)(p,(y), y), where this
function changes signs. Now,
(o2II)(p,(y), y) = exp (- Jo 1 [jt(y)Q(s)a'(8(s; jt(y), y)) + g'(8(s; jt(y), Y))]ds),
thus the integral carries the information about stability.
Remark 2. Floquet theory and the principle of linearized stability tell us that for given
(Jl, y) E ~ 8(·; Jl, y) is an asymptotically stable (unstable) periodic orbit provided that
(02IT)(jt(y), y) < 1 (> 1), which corresponds to jt'(y) > 0 « 0). Thus, forward bending
segments of the curve y f-+ (P,(y), y) yield climates, backward bending segments do not,
i.e. the situation is quite the same as for (4).
Let us now see how the fact that Q has zeroes may affect the a priori estimate from
below.
Example. We consider (8) with constant albedo and a Stefan-Boltzmann like outgoing
radiation flux. Thus, we obtain the following initial value problem:
(11)
i+0"(Z)Z4 = Jlq(t)
t> 0,
We assume
{
9 : Y f-+ 0"(y)y4 E COO(R+), g' I (0,00) positive, ,0"' has compact support in
(0,00),Q.:= infO" > 0, q E C 2 (R) I-periodic and nonnegative,
:3 0 < T{ < T; < 1 with supp(q I [0,1]) = [0,1] \ (Tt, Tn.
We are going to show that lim inft~oo z( t) S; [3Q.( T; - Tnr1/3, whenever z solves (11).
One obtains the above estimate by comparing z in (11) with the solution ( of
t > 0,
((0) = z(O),
for t E R+. Thus, b> 0 yields (01IT)(Jl,Y) = (028)(1; Jl,Y) > 0, hence for each Y E R+
there exists exactly one p,(y) with IT(jt(y),y) = y. The implicit function theorem shows
jt E C 2 ((0, 00), (0, 00)) and (10) implies jt to be continuous at y = 0 with value o.
Again, turning points of ~ are those (p,(y), y), where jt(y) a strict local extrema of
jt. We obtain by implicit differentation that
"() 1-(~IT)(jt(y),y)
Jl y = (01 IT)(p,(y), y)
Y E (0,00),
hence turning points of ~ occur at those zeroes of y f-+ 1 - (02II)(p,(y), y), where this
function changes signs. Now,
(o2II)(p,(y), y) = exp (- Jo 1 [jt(y)Q(s)a'(8(s; jt(y), y)) + g'(8(s; jt(y), Y))]ds),
thus the integral carries the information about stability.
Remark 2. Floquet theory and the principle of linearized stability tell us that for given
(Jl, y) E ~ 8(·; Jl, y) is an asymptotically stable (unstable) periodic orbit provided that
(02IT)(jt(y), y) < 1 (> 1), which corresponds to jt'(y) > 0 « 0). Thus, forward bending
segments of the curve y f-+ (P,(y), y) yield climates, backward bending segments do not,
i.e. the situation is quite the same as for (4).
Let us now see how the fact that Q has zeroes may affect the a priori estimate from
below.
Example. We consider (8) with constant albedo and a Stefan-Boltzmann like outgoing
radiation flux. Thus, we obtain the following initial value problem:
(11)
i+0"(Z)Z4 = Jlq(t)
t> 0,
We assume
{
9 : Y f-+ 0"(y)y4 E COO(R+), g' I (0,00) positive, ,0"' has compact support in
(0,00),Q.:= infO" > 0, q E C 2 (R) I-periodic and nonnegative,
:3 0 < T{ < T; < 1 with supp(q I [0,1]) = [0,1] \ (Tt, Tn.
We are going to show that lim inft~oo z( t) S; [3Q.( T; - Tnr1/3, whenever z solves (11).
One obtains the above estimate by comparing z in (11) with the solution ( of
t > 0,
((0) = z(O),