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In this paper a family of non-autonomous scalar parabolic PDEs over a generalcompact and connected flow is considered. The existence or not of aneighbourhood of zero where the problems are linear has an influence on themethods used and on the dynamics of the induced skew-product semiflow. That iswhy two cases are distinguished: linear-dissipative and purely dissipativeproblems. In both cases, the structure of the global and pullback attractors isstudied using principal spectral theory. Besides, in the purely dissipativesetting, a simple condition is given, involving both the underlying lineardynamics and some properties of the nonlinear term, to determine the nontrivialsections of the attractor.

Non-autonomous scalar linear-dissipative and purely dissipative parabolic PDEs over a compact base flow