Geodesic
A mathematical model of heat transmission in essentially nonlinear conjugate media
Matematičeskoe modelirovanie, Tome 2 (1990) no. 6, pp. 40-54 Cet article a éte moissonné depuis la source Math-Net.Ru

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Heat transmission in conjugate systems (cubes, cylinders, spheres) with travelling boundary line is investigated on the basis of the heat-conductivity nonlinear equation with a heat source obtained from the solution of an electrodynamic problem in a nonlinear medium whose dielectric penetrability depends on the field according to the law $$ \varepsilon_i=\varepsilon_{0i}-|\alpha_i|E_i^2,\qquad i=1,2. $$ The analysis carried out points out some nontrivial effects accompanying the heat transmission: the appearance of adiabatic surfaces, emergence of solition solutions and the conditions with sharpening in self-focusing media. All the considered effects are essentially defined by the values of $\alpha_i$ nonlinear parameters, nature of dependences of dielectric penetrabilities on the temperature, and the character of movement of the boundary surface. 
@article{MM_1990_2_6_a4,
     author = {L. A. Uvarova and V. K. Fedyanin},
     title = {A~mathematical model of heat transmission in essentially nonlinear conjugate media},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {40--54},
     year = {1990},
     volume = {2},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_1990_2_6_a4/}
}
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L. A. Uvarova; V. K. Fedyanin. A mathematical model of heat transmission in essentially nonlinear conjugate media. Matematičeskoe modelirovanie, Tome 2 (1990) no. 6, pp. 40-54. http://geodesic.mathdoc.fr/item/MM_1990_2_6_a4/