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 l) 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.w Introduction. The structure of parabolic equations entails a certain "disparity" of the independent variable t, corresponding to the time, when comparison is made with the spatial variables x = (x 1 ..... x n) . In particular, this finds its expression in the fact that when methods used to investigate the smoothness with respect to x of the solutions of such equations are applied to investigating smoothness with respect to t, they usually lead to the necessity of introducing certain very objectionable conditions (not fundamentally called for) on the functions defining the equation, especially when the latter is nonlinear. It is then natural, in order to obtain more precise results concerning smoothness of the solutions of parabolic equations, to apply methods which take into account the specific nature of the "disparity" mentioned above between the var{ables t and x, one of the features of this disparity being a definite "subordination" of the properties of the solutions with respect to t to their properties with respect to x.A basic aim of the present paper consists in proving that the moduli of continuity with respect to t (in the spaces C and L1), for bounded solutions of certain classes of parabolic equations, may be estimated in terms of corresponding moduli of continuity with respect to x, under minimal (in a definite sense) assumptions on the functions defining the equations; the equations considered are, in general, nonlinear and degenerate. We devote w167 2 and 4 for this purpose. Analogous results for the case of powers of the moduli of continuity (in C) for the solutions of linear parabolic equations were obtained in [1]; the method used for proving Theorem 2 in w 4 is based on ideas used in [2].In w 3 we consider bounded solutions of the Cauchy problem for the unsteady filtration equationwith the initial conditionUsing Theorem I of w an estimate is obtained of the modulus of continuity with respect to t (in C) for the solution of problem (1), (2); this estimate supplements a recent result of D. Aronson [3], a result whose discussion at O. A. Oleinik's seminar at Moscow State University stimulated to a considerable extent the writing of w w 2 and 3 of the present paper.
We present a model for microwave photoconductivity in two-dimensional electron systems (2DESs) in a magnetic field at microwave frequencies lower than the electron cyclotron frequency when the intra-Landau level (LL) transitions dominate. Using this model, we explain the effect of the decrease in the 2DES dissipative conductivity (and resistivity) and smearing of its Shubnikov-de Haas oscillations caused by microwave radiation observed recently. The model invokes the concept of suppression of elastic impurity scattering of electrons by the microwave electric field. We calculated the dependence of the 2DES conductivity associated with intra-LL transitions as a function of the radiation and cyclotron frequencies and microwave power. We take into account the effect of plasma reflection of microwaves from 2DES resulting in a distinction between the ac microwave electric field acting on electrons and that generated by a microwave source. The obtained dependences are consistent with the results of recent experimental observations.
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