279
C+(M), we see that the stationary solutions of (17) coincide with those of (1) in case
that a(x,y) := a(x,y,y). Thus, the stability properties of the equilibria are really at
issue here.
Let us reformulate the "basic hypotheses" of Section 2. in a form suitable for a
setting described by (17).
Basic hypotheses.
(H"O) M 2-dimensional, compact, oriented Riemannian manifold without boundary,
T E (0,00);
(H"l) Q,c,k E C 2 (M) positive; f3 E COO([-T,OJ), f3(-T) = 0, f3(s) > 0 for s E (-T, 0]
and J~Tf3(s)ds = 1;
(H"2) a E C 2 (M x R+ x R+), infO' > 0, sUpO' < I,
03a(X,y,Z) < 0 for x E M,y E R+ and z > 0,
lim y J(gradO')(x,·, .)(y, y)J = 0 uniformly in x E M;
y-+oo
(H"3) g E C 2 (R+), g(O) = 0, g'(y) > 0 for y E (0,00), g(y) -+ 00 as y -+ 00,
-.- g(y)
ll'm J(grada)(x,·,·)(y,y)J = 0
hm -,---( ) < 00,
( )
y-+ooyg y
y-+o+
g' y
uniformly for x EM.
Remark 4- A setup O'(x, y, z) = al(x, y)+0'2(X, z) is a simple example for an 0' in (H"2).
The total albedo results in that case from a superposition of the albedo due to clouds,
water, seasonal phenomena and so forth, which is modeled by aI, with the albedo due
to the expansion or retreat of the continental ice-sheets or mountain glaciers modeled by
0'2. This approach provides the reason for our assumption that 0' is (strictly) decreasing
in its third argument. Note that [Bhattacharya et al. (1982)] links the total albedo to
the chosen long-term mean of temperature and thereby suppresses the short time-scales.
Therefore it would be problematic in such a setting to require a to be nonincreasing
with respect to that temperature mean, e.g. in view of the cloud albl,do.
Considering the delay term J~Tf3(s)u(t + s,x)d.s in (17) we need to prescribe a
temperature history {) E C+([-T, 0] x M) as initial condition for (17), i.e. one demands
C+(M), we see that the stationary solutions of (17) coincide with those of (1) in case
that a(x,y) := a(x,y,y). Thus, the stability properties of the equilibria are really at
issue here.
Let us reformulate the "basic hypotheses" of Section 2. in a form suitable for a
setting described by (17).
Basic hypotheses.
(H"O) M 2-dimensional, compact, oriented Riemannian manifold without boundary,
T E (0,00);
(H"l) Q,c,k E C 2 (M) positive; f3 E COO([-T,OJ), f3(-T) = 0, f3(s) > 0 for s E (-T, 0]
and J~Tf3(s)ds = 1;
(H"2) a E C 2 (M x R+ x R+), infO' > 0, sUpO' < I,
03a(X,y,Z) < 0 for x E M,y E R+ and z > 0,
lim y J(gradO')(x,·, .)(y, y)J = 0 uniformly in x E M;
y-+oo
(H"3) g E C 2 (R+), g(O) = 0, g'(y) > 0 for y E (0,00), g(y) -+ 00 as y -+ 00,
-.- g(y)
ll'm J(grada)(x,·,·)(y,y)J = 0
hm -,---( ) < 00,
( )
y-+ooyg y
y-+o+
g' y
uniformly for x EM.
Remark 4- A setup O'(x, y, z) = al(x, y)+0'2(X, z) is a simple example for an 0' in (H"2).
The total albedo results in that case from a superposition of the albedo due to clouds,
water, seasonal phenomena and so forth, which is modeled by aI, with the albedo due
to the expansion or retreat of the continental ice-sheets or mountain glaciers modeled by
0'2. This approach provides the reason for our assumption that 0' is (strictly) decreasing
in its third argument. Note that [Bhattacharya et al. (1982)] links the total albedo to
the chosen long-term mean of temperature and thereby suppresses the short time-scales.
Therefore it would be problematic in such a setting to require a to be nonincreasing
with respect to that temperature mean, e.g. in view of the cloud albl,do.
Considering the delay term J~Tf3(s)u(t + s,x)d.s in (17) we need to prescribe a
temperature history {) E C+([-T, 0] x M) as initial condition for (17), i.e. one demands