Geodesic
A priori smoothness of solutions for number of equations of a changing type
Matematičeskoe modelirovanie, Tome 2 (1990) no. 9, pp. 145-153 
M. M. Lavrent'ev (Jn.). A priori smoothness of solutions for number of equations of a changing type. Matematičeskoe modelirovanie, Tome 2 (1990) no. 9, pp. 145-153. http://geodesic.mathdoc.fr/item/MM_1990_2_9_a12/
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     author = {M. M. Lavrent'ev (Jn.)},
     title = {A~priori smoothness of solutions for number of equations of a~changing type},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {145--153},
     year = {1990},
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     url = {http://geodesic.mathdoc.fr/item/MM_1990_2_9_a12/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

For number of non-linear equations of the type $$ U_t=a''(U_x)U_{xx}+2\mu U U_x, $$ with sign changing function $a''(\xi)$ ($a''(\xi)\geqslant\delta>0$, $|\xi|\geqslant N$) a priori estimation $\|u_x\|_{W_2^{1,1}}$ for smooth solutions in obtained. Different form the previous investigations the case of $\mu\ne0$ and the more general form of the function $a$ are considered,Connection is marked of the problem considered with so-called Cahn–Hilliard equation by which the phase separation in the melts can be simulated.