Nonlinear thermal evolution in an inhomogeneous medium

Various simple transport models of electron temperature in a confined plasma are reducible to the quasilinear equation ρ(x)ut = [c(x)unx]x +A(x)us, −1<x<1, u(±1) = 0. u is the temperature, ρ(x) the density, and c = g[ ρ(x)] the density-dependent part of the thermal diffusion. ρ(x) and c(x) may vanish at the plasma edge, rendering the problem singular. The temporal behavior depends critically on the boundedness of R = ℱ+1−1 c−1(x) dx. If R<∞ then in the absence of heat sources, A≡0, every initially given state u(x, 0) evolves toward an algebraically decaying, universal space- time separable solution. Its existence and uniqueness is proved. The method developed in this work may be used to show the equilibrization of the solution in the presence of a heat source of the form A(x)us, s<n, ρ(x)≳0. On the other hand, if R = ∞ and A = 0 then the system becomes isothermalized: u→ū = ℱ+1−1 u(x, 0) ρ(x) dx/ℱ1−1 ρ(x) dx≳0. In such a case addition of heat sources will cause a thermal explosion.