On solvability of parabolic equations with essentially nonlinear differential-difference operators

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