On Solvability of Parabolic Equations with Essentially Nonlinear Differential-Difference Operators

We consider the first mixed boundary value problem for a nonlineardifferential-difference parabolic equation. We give some sufficient conditions for the nonlinear differential-differenceoperator to be radially continuous and coercive as well as has the property of (V,W)-semibounded variation(in this case we providethe algebraic condition of strong ellipticity for an essentiallynonlinear differential-difference operator).We also justify the existence theorems for a generalized solution.