On Comparison Theorems in Nonlinear Evolution Equation

We first introduce several notations.

R^d = d-dimensional Euclidean space,

B^d = Borel σ algebra on R ^d,

C(R^d ) = the family of all continuous functions on R ^d,

C² (R ²) = the family of all twice differentiable functions on R ^d,

B(R^d ) = the family of all bounded Borel measurable functions on R^d ,

bpC(R^d ) = the family of all bounded positive continuous functions on R^d ,

M(R^d ) = the set of all finite measure on R^d.