  We show that for any uniformly parabolic fully nonlinear second-order
equation with bounded measurable "coefficients" and bounded "free" term in the
whole space or in any cylindrical smooth domain with smooth boundary data one
can find an approximating equation which has a continuous solution with the
first and the second spatial derivatives under control: bounded in the case of
the whole space and locally bounded in case of equations in cylinders. The
approximating equation is constructed in such a way that it modifies the
original one only for large values of the second spatial derivatives of the
unknown function. This is different from a previous work of Hongjie Dong and
the author where the modification was done for large values of the unknown
function and its spatial derivatives.
