The purpose of this chapter is to furnish diagrams that illustrate the properties
stated in Theorems 1, 2, 3 and 4 as a function of the parameters p and q.
While the figure below suggest that the curves are monotone this is yet to be
proven. The reader is invited to [5, 6] where the geometry of the bifurcation curves
is studied when p = (N + 2)/(N − 2)
To interpret the physical significance of the bifurcation diagrams is convenient
to refer the values of p and q to the corresponding values of the parameter k, m,
and n. For example, in a plasma for which k = 5/2, m = −1, n = −3/2 (see
[11, page 179]) we have p = −2/7 and q = −3/7. This case correspond to Figure
5. In order to simplify our discussion we will illustrate the interpretation of the
diagrams only in the case that k + 1 0. Similar interpretations can be obtained
when k + 1 0.
We also recall that the parameter d is proportional to the temperature in the
center of the configuration, and λ to its radius.
For values of k, m, n leading to 1 ≤ q p (N + 2)/(N − 2) Figure 1 shows
that there cannot be steady states with absorbing zones when the temperature
at the center is very large. Also, those states with very large temperature in the
center are only possible for small radius of the ball. The number of absorbing zones
increases with the radius of the ball.
In the case that 1 ≤ q p, p ≥
, Figure 2 shows that for a given radius
the number of absorbing zones is bounded. For either thermal equilibrium or no
flux at the boundary, high temperatures at the center are only possible when the
radius of the star is near the values μj/(p − q) or ν/(p − q), respectively. For very
low temperatures steady states are possible only in very large balls.
For values of of k, m, n leading to −1 q 1 p (N + 2)/(N − 2) we see
that there cannot be steady states with absorbing zones when the temperature at
the center is very large. Determining the value of D becomes then an important
practical problem to be addressed. Figure 3 also shows that the number of absorbing
zones in bounded in terms of the radius of the ball.
Figure 4 proves that for a given radius the number of radiating and absorbing
zones is bounded. Equivalently, in order to have a steady state with a large number
of radiating an absorbing zones one needs a very larger region. Also there are values
of the radii for which the temperature at the center may be very large while the
number of absorbing zones stays bounded.
If −1 q p 1 we see that steady states are possible with very low
temperatures and small radii. Also Figure 5 shows that steady states with high
temperatures at the center are only possible when the radius is large. Moreover,
the threshold D determines an upper bound for temperatures anywhere in the star
in the presence of absorbing zones. This case includes n ∈ (0, 1) for a plasma in