  Results on unconditional convergence in the Maximum norm for ADI-type
methods, such as the Douglas method, applied to the time integration of
semilinear parabolic problems are quite difficult to get, mainly when the
number of space dimensions $m$ is greater than two. Such a result is obtained
here under quite general conditions on the PDE problem in case that
time-independent Dirichlet boundary conditions are imposed. To get these
bounds, a theorem that guarantees, in some sense, power-boundeness of the
stability function independently of both the space and time resolutions is
proved.
