On the existence of solution of some problems for nonlinear loaded parabolic equations with Cauchy data

Two problems are considered in this paper. First problem is the Cauchy problem for a two-dimensional loaded parabolic equation with coefficients dependent on unknown function and its derivatives. Second problem is the Cauchy problem for one-dimensional equation of the Burgers-type. The sufficient conditions of the existence of solutions of these problems in classes of smooth bounded functions are presented in the paper. The method of weak approximation is used for the purpose of obtaining the proof.