Results concerning the nature of the continuity of solutions of parabolic equations and some of their applications

Theorems are established which deal with the relationship between the moduli of continuity with respect to $x$ and with respect to $t$ (in the spaces $C$ and $L_1$) for the solutions of certain classes of parabolic equations, which are, in general, nonlinear and degenerate. These theorems are then applied to an investigation of the Cauchy problem for quasilinear equations of the first and second order.