TY - JOUR

T1 - Nonlinear thermal evolution in an inhomogeneous medium

AU - Kamin, Shoshana

AU - Rosenau, Philip

PY - 1981

Y1 - 1981

N2 - Various simple transport models of electron temperature in a confined plasma are reducible to the quasilinear equation ρ(x)ut = [c(x)uxn]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 = ∫ -1 +1c-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, γ(x) > 0. On the other hand, if R = ∞ and A = 0 then the system becomes isothermalized: u→ū = ∫-1+1u(x, 0)ρ(x) dx/∫1-1 ρ(x) dx > 0. In such a case addition of heat sources will cause a thermal explosion.

AB - Various simple transport models of electron temperature in a confined plasma are reducible to the quasilinear equation ρ(x)ut = [c(x)uxn]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 = ∫ -1 +1c-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, γ(x) > 0. On the other hand, if R = ∞ and A = 0 then the system becomes isothermalized: u→ū = ∫-1+1u(x, 0)ρ(x) dx/∫1-1 ρ(x) dx > 0. In such a case addition of heat sources will cause a thermal explosion.

UR - http://www.scopus.com/inward/record.url?scp=0013374879&partnerID=8YFLogxK

U2 - 10.1063/1.525506

DO - 10.1063/1.525506

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AN - SCOPUS:0013374879

SN - 0022-2488

VL - 23

SP - 1385

EP - 1390

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 7

ER -