  Systems of non-autonomous parabolic partial differential equations over a
bounded domain with nonlinear term of Carath\'eodory type are considered.
Appropriate topologies on sets of Lipschitz Carath\'eodory maps are defined in
order to have a continuous dependence of the mild solutions with respect to the
variation of both the nonlinear term and the initial conditions, under
different assumptions on the bound-maps of the nonlinearities.
