On the Solvability of Nonlinear Parabolic Functional-Differential Equations with Shifts in the Spatial Variables

The first mixed boundary value problem for a nonlinear functional-differential equation of parabolic type with shifts in the spatial variables is considered. Sufficient conditions are proved under which a nonlinear differential-difference operator is demicontinuous, coercive, and pseudomonotone on the domain of the operator $\partial_t$. Based on these properties, existence theorems for a generalized solution are proved.