266
which intersect transversally at the bifurcation point. The bifurcation is global in the
sense of Rabinowitz with at least another bifurcation point of the same type on the bifurcation branch. Of course, near the bifurcation point there are only unstable solutions.
She also studied the stability of these bifurcation points under small perturbations. Her
findings suggest that the scenario described by (H4) and (H5) above is the most likely
one to expect for (1).
It is worthwhile for future considerations to isolate the key features that lead to
S-shapedness in the proof outlined above:
• a priori estimates;
• unique solvability for small and large fl;
• existence of a principal eigenvalue;
• validity of a principle of linearized stability.
5. Time Periodic Forcing.
So far we have qualitatively studied the dependence of the earth's climates on the
so-called solar constant in the framework of a I-layer energy balance model without
seasonal forcing. In order to account for seasonal effects, one has to replace Q=Q(x) in
(1) by an in t I-periodic function Q=Q(t,x), which represents the seasonally averaged
incoming solar radiation flux. One may think of, say, a mean
4 5 if;
Q(t,x)=- L Q(t+s+j,x)ds,
11 j=-5 -i
where t :::: 0 denotes time in years and x E 8 2 stands for a surface point of the earth.
Actually, Q(t,·) means the incoming flux at the top of the atmosphere, which can be
described though by surface coordinates via radial retraction.
Of course, the climatological indicator of our model, the temperature u, is now understood as correspondently averaged, and stable, in time I-periodic solutions represent
the earth's climates in this new setting. The modified model equation reads then as
(7) c(x)Otu(t,x) - div(k(.) gradu(t, ·))(x) = p Q(t,x)[1 - a(x,u(t,x))] - g(u(t,x)),
which intersect transversally at the bifurcation point. The bifurcation is global in the
sense of Rabinowitz with at least another bifurcation point of the same type on the bifurcation branch. Of course, near the bifurcation point there are only unstable solutions.
She also studied the stability of these bifurcation points under small perturbations. Her
findings suggest that the scenario described by (H4) and (H5) above is the most likely
one to expect for (1).
It is worthwhile for future considerations to isolate the key features that lead to
S-shapedness in the proof outlined above:
• a priori estimates;
• unique solvability for small and large fl;
• existence of a principal eigenvalue;
• validity of a principle of linearized stability.
5. Time Periodic Forcing.
So far we have qualitatively studied the dependence of the earth's climates on the
so-called solar constant in the framework of a I-layer energy balance model without
seasonal forcing. In order to account for seasonal effects, one has to replace Q=Q(x) in
(1) by an in t I-periodic function Q=Q(t,x), which represents the seasonally averaged
incoming solar radiation flux. One may think of, say, a mean
4 5 if;
Q(t,x)=- L Q(t+s+j,x)ds,
11 j=-5 -i
where t :::: 0 denotes time in years and x E 8 2 stands for a surface point of the earth.
Actually, Q(t,·) means the incoming flux at the top of the atmosphere, which can be
described though by surface coordinates via radial retraction.
Of course, the climatological indicator of our model, the temperature u, is now understood as correspondently averaged, and stable, in time I-periodic solutions represent
the earth's climates in this new setting. The modified model equation reads then as
(7) c(x)Otu(t,x) - div(k(.) gradu(t, ·))(x) = p Q(t,x)[1 - a(x,u(t,x))] - g(u(t,x)),