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solution semiflow is gradient-like. Thus each solution trajectory u( t,·; fl., 19) converges
as t ----4 00 to the set of equilibria w given by
(3)
-div (k grad w )(x) == fl. Q(x)[l - o:(x, w(x ))] - g( w(x )).
Considering the climates as represented by the persisting pattern in such a model we
should identify them with the stable stationary solutions of (3). Note that it is an easy
consequence of the Absorbing Property mentioned above that the set of equilibria is
bounded in C(M) for fixed fl. E R+. Let us summarize our discussion:
Proposition 1. Let (HO) - (H3) be satisfied and fl. E R+. Tben tbe initial value
problem (1), u(O,·) == 19 E C+(M), bas a unique global nonnegative solution u(·,·; fl., 19) :
R+ xM ----4 R+. Tbis solution approacbes tbe set of stationary solutions of (3) as t ----4 00.
The situation for multi-layer energy balance models is more complicated as far as the
dynamics is concerned. We refer to [Hetzer/Schmidt (1990, 1992)] for a mathematical
analysis of reaction-diffusion systems arising in that context. In particular, the gradientlike structure is lost, and we can hope at best for statements that "most" solution
trajectories approach the set of equilibria (cf. [Hirsch (1988)] for the theory of order
preserving semifiows.)
4. The Concept of S-shapedness.
As pointed out before we are interested in the stationary solutions of (3) and
their dependence on fl.. It is therefore convenient to introduce the "solution set"
<5 == {(Il,w) E R+ x C+(M) : (Il,w) solves (3)} and its principal branch~, which
is the connectivity component of <5 containing (0,0).
Let us begin by discussing a simple example that fits into a first course in differential
equations.
Assume one wants to calculate the earth's global temperature average 6 from the
radiation flux budget. Employing the same notations for albedo, incoming solar radiation flux and outgoing terrestrial radiation flux as before, but overlined in order to
indicate the global average, we obtain
(4)
e(t) == IlQ[l - a(6(t))] - g(0(t))
t > O.
solution semiflow is gradient-like. Thus each solution trajectory u( t,·; fl., 19) converges
as t ----4 00 to the set of equilibria w given by
(3)
-div (k grad w )(x) == fl. Q(x)[l - o:(x, w(x ))] - g( w(x )).
Considering the climates as represented by the persisting pattern in such a model we
should identify them with the stable stationary solutions of (3). Note that it is an easy
consequence of the Absorbing Property mentioned above that the set of equilibria is
bounded in C(M) for fixed fl. E R+. Let us summarize our discussion:
Proposition 1. Let (HO) - (H3) be satisfied and fl. E R+. Tben tbe initial value
problem (1), u(O,·) == 19 E C+(M), bas a unique global nonnegative solution u(·,·; fl., 19) :
R+ xM ----4 R+. Tbis solution approacbes tbe set of stationary solutions of (3) as t ----4 00.
The situation for multi-layer energy balance models is more complicated as far as the
dynamics is concerned. We refer to [Hetzer/Schmidt (1990, 1992)] for a mathematical
analysis of reaction-diffusion systems arising in that context. In particular, the gradientlike structure is lost, and we can hope at best for statements that "most" solution
trajectories approach the set of equilibria (cf. [Hirsch (1988)] for the theory of order
preserving semifiows.)
4. The Concept of S-shapedness.
As pointed out before we are interested in the stationary solutions of (3) and
their dependence on fl.. It is therefore convenient to introduce the "solution set"
<5 == {(Il,w) E R+ x C+(M) : (Il,w) solves (3)} and its principal branch~, which
is the connectivity component of <5 containing (0,0).
Let us begin by discussing a simple example that fits into a first course in differential
equations.
Assume one wants to calculate the earth's global temperature average 6 from the
radiation flux budget. Employing the same notations for albedo, incoming solar radiation flux and outgoing terrestrial radiation flux as before, but overlined in order to
indicate the global average, we obtain
(4)
e(t) == IlQ[l - a(6(t))] - g(0(t))
t > O.