Functional differential inequalities with partial derivatives

Initial problems for Hamilton-Jacobi functional differential equations and initial boundary value problems of the Dirichlet type for parabolic equations are considered. It is proved that classical solutions of functional differential equations can be estimated by solutions of initial problems for ordinary functional differential equations. Theorems on the uniqueness of solutions are obtained as consequences of comparison results. A method of differential inequalities is used. Here, the involved operators do not satisfy the Volterra condition.