Introduction

The modeling of thermal structures, e.g. plasmas, is determined by three basic

energy transport mechanisms: heat diffusion, heat generation, and energy radiation.

For static configurations, in which the transfer of energy takes place entirely by

thermal conduction, the equations of conservation of energy of a thermal structure

confined in a region Ω ⊂

RN

can be written as

(0.1) ρc

∂u

∂t

= ∇[κ(ρ, u)∇u] + Γ(ρ, u) − Λ(ρ, u) ,

where u :

¯

Ω → (0, ∞) denotes the temperature, ρ :

¯

Ω → (0, ∞) denotes the density

of the thermal structure, c denotes the specific heat per unit of mass, κ(ρ, u) is

the heat diffusion coeﬃcient, Γ(ρ, u) is the heat generated (usually due to nuclear

fusion) per unit volume and time, Λ(ρ, u) is the heat radiated per unit volume and

time.

Important physical situations correspond to the case where

(0.2) κ =

κ0uk

, Γ =

Γ0um,

Λ =

Λ0un

,

where κ0, k, Γ0, m, Λ0, and n are given constants. For example, in the absence of

magnetic fields k can be taken to be 5/2, whereas k = −5/2 when the diffusion takes

place perpendicular to a magnetic field. For constant heating per unit of volume

m = 0 and for constant heating per unit of mass m = 1. Values of n ranging from

-2.84 to 5.17 have been found to be adequate for various experimental cases ([13]).

For constant density and assuming that Ω = B(0, R), a ball of radius R, equa-

tion (0.1) can be written in dimensionless form as

(0.3)

∂u

∂t

=

∇(uk∇u)

+

λ(um

−

un)

in B(0, 1) ×

R+,

with x → x/R, u → u/T , T = (Γ0/Λ0)1/(m−n), t →

tk0T

k

R2ρc

and λ =

Γ0

k0

R2T m−k−1.

The parameter T is the temperature of thermal equilibrium (energy radiation =

energy generation). We study (0.3) under the initial-boundary conditions

(0.4) u(·, 0) = u0 in B(0, 1), u(·, t) = 1 in ∂B(0, 1) ×

R+,

or

(0.5) u(·, 0) = u0 in B(0, 1),

∂u(·, t)

∂ν

= 0 in ∂B(0, 1) ×

R+.

Radial steady states of (0.3) are then solutions to

(0.6)

1

rN−1

∂

∂r

(rN−1uk

∂u

∂r

) +

λ(um

−

un)

= 0,

and the boundary conditions in (0.4) and (0.5) become, respectively,

(0.7) u(1) = 1 thermal equilibrium,

1