Critical conditions and comparison theorems for difference solutions of nonlinear parabolic equations

Conditions for special pointwise estimates of difference solutions, i.e., criticality conditions, to exist, are established, and are used to prove comparison theorems for the solutions of implicit difference schemes for various non-linear parabolic equations, i.e., the conditions are found, under which one solution is majorized by another at all nodes of the space-time mesh. The results reveal that the solutions of implicit schemes display important properties of “monotonicity”, inherent in differential problems for non-linear parabolic equations.