Geodesic
Explicit solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity
Bollettino della Unione matematica italiana, Série 8, 9B (2006) no. 1, pp. 79-99

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We study a one-phase Stefan problem for a semi-infinite material with temperature-dependent thermal conductivity with a constant temperature or a heat flux condition of the type $-q_0/\sqrt{t}$ ($q_0 > 0$) at the fixed face $x=0$. We obtain in both cases sufficient conditions for data in order to have a parametric representation of the solution of the similarity type for $t\geq t_0>0$ with $t_0$ an arbitrary positive time. These explicit solutions are obtained through the unique solution of an integral equation with the time as a parameter. 
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     author = {Natale, Mar{\'\i}a F. and Tarzia, Domingo A.},
     title = {Explicit solutions for a one-phase {Stefan} problem with temperature-dependent thermal conductivity},
     journal = {Bollettino della Unione matematica italiana},
     pages = {79--99},
     publisher = {mathdoc},
     volume = {Ser. 8, 9B},
     number = {1},
     year = {2006},
     zbl = {1118.80005},
     mrnumber = {MR2204902},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2006_8_9B_1_a4/}
}
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Natale, María F.; Tarzia, Domingo A. Explicit solutions for a one-phase Stefan problem with temperature-dependent thermal conductivity. Bollettino della Unione matematica italiana, Série 8, 9B (2006) no. 1, pp. 79-99. http://geodesic.mathdoc.fr/item/BUMI_2006_8_9B_1_a4/