  In this paper a family of non-autonomous scalar parabolic PDEs over a general
compact and connected flow is considered. The existence or not of a
neighbourhood of zero where the problems are linear has an influence on the
methods used and on the dynamics of the induced skew-product semiflow. That is
why two cases are distinguished: linear-dissipative and purely dissipative
problems. In both cases, the structure of the global and pullback attractors is
studied using principal spectral theory. Besides, in the purely dissipative
setting, a simple condition is given, involving both the underlying linear
dynamics and some properties of the nonlinear term, to determine the nontrivial
sections of the attractor.
