Geodesic
On a problem with free boundary for parabolic equations
Sbornik. Mathematics, Tome 43 (1982) no. 4, pp. 473-484 
A. M. Meirmanov. On a problem with free boundary for parabolic equations. Sbornik. Mathematics, Tome 43 (1982) no. 4, pp. 473-484. http://geodesic.mathdoc.fr/item/SM_1982_43_4_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

This paper considers the problem of determining a solution of the parabolic equation $$ L\theta\equiv D_t\theta-\sum^2_{i,j=1}D_i(a_{ij}(x,t,\theta)\cdot D_j\theta)+a(x,t,\theta,D\theta)=0 $$ and the boundary of the two-dimensional region in which a solution of the equation is sought in the case where on the free boundary the value of the desired function and the additional condition $$ \sum^2_{i,j=1}a_{ij}D_i\theta\cdot D_j\theta=g(x,t) $$ are satisfied. For this problem a theorem asserting the existence of a smooth solution on a small time interval is proved. If $L\theta=0$ is the heat equation, then the solution exists on any time interval, and it is unique. Bibliography: 7 titles.

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[3] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

[4] Meirmanov A. M., “O klassicheskom reshenii mnogomernoi zadachi Stefana dlya kvazilineinykh parabolicheskikh uravnenii”, Matem. sb., 112 (154) (1980), 170–192 | MR | Zbl

[5] Miranda K., Uravneniya s chastnymi proizvodnymi ellipticheskogo tipa, IL, M., 1967

[6] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[7] Nguen Din Ni, “Ob odnoi zadache so svobodnoi granitsei dlya parabolicheskogo uravneniya”, Vestn. MGU, matem., mekhan., 1966, no. 2, 40–54 ; No 5, 51–62 | Zbl