A priori smoothness of solutions for number of equations of a changing type

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.