2 A. CASTRO AND V. PADR

´

ON

or

(0.8) u (1) = 0 no flux.

For k = −1 taking v = uk+1 we see that the steady configurations of the

thermal structure are given by solutions to the semilinear equation

(0.9) v (r) +

N − 1

r

v (r) +

λ(vp(r)

−

vq(r))

= 0, u (0) = 0, r 0.

with λ → λ(k + 1), p =

m

k+1

, q =

n

k+1

if k + 1 0; and λ → −λ(k + 1), p =

n

k+1

,

q =

m

k+1

if k + 1 0.

If k = −1, setting v = ln(u), equation (0.3) is equivalent to

(0.10) v (r) +

N − 1

r

v (r) +

λ(emv(r)

−

env(r))

= 0, u (0) = 0, r 0.

This case requires independent analysis and is presented in Chapter 6.

We classify (see Theorems 1, 2, 3 and 4) the positive solutions to the equation

(0.9) for N ≥ 2, q p and λ 0 subject to either (0.7) or (0.8). We emphasize

that p or q may exceed the critical exponent

2∗

− 1 = (N + 2)/(N − 2).

In the conservative case, N = 1, the results are similar and can be obtained

directly from the phase plane analysis. This is presented in Appendix 1.

We also provide information on the stability-unstability of the radial steady

states (see Theorems 6 and 8). The reader is referred to [10, 11, 12, 13, 14] for

additional information on the stability of the steady state solution given by the

thermal equilibrium temperature.

Remark 1. Our results answer questions such as:

• What is the radius of a spherical star that supports a radial tempera-

ture distribution with a given number of hot (radiating, u 1) and cold

(absorbing, u 1) zones?

• What values of parameters k, m, n allow stable steady states with arbi-

trarily large temperature in the center?

• What values of k, m, n allow steady states with arbitrarily large temper-

ature and absorbing zones?

• When is a given temperature distribution stable?

Remark 2. Assuming the thermal structure to have constant pressure, in

dimensionless form (0.1) becomes

(0.11)

∂u

∂t

= u

∇(uk∇u)

+

λ(um

−

un)

in B(0, 1) ×

R+.

and its radial steady states are also solutions to (0.9). Thus our classification of

radial steady states extends to the constant pressure case.

To study the problem (0.9) subject to (0.7) or to (0.8) we analyze the positive

solutions to the initial value problem

⎧

⎨

⎩

v (t) +

N−1

t

v (t) +

λ(vp(t)

−

vq(t))

= 0, t 0

v(0) = d, v (0) = 0.

(0.12)

We will denote by v(·, λ, d) the solution to the initial value problem (0.12). This

function satisfies the re-scaling property

(0.13) v(λt, 1, d) = v(t,

λ2,

d).