On viscosity solutions for non-totally parabolic fully nonlinear equations

It is shown, how to prove global unique solvability of the first initial-boundary value problem in the class of continuous viscosity solutions for some classes of equations $-u_t+\mathcal F(u_x,u_{xx})=g(x,t,u_x)$ wit $\mathcal F(p,A)$ determined and elliptic only on some nonlinear subsets of values of arguments $(p,A)$. For this purpose we use the technic developed in the theory of viscosity solutions for degenerate elliptic equations. Bibl. 12 titles.