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

The paper’s Theorem 5.5 proves that in the purely dissipative setting, c(t,p0)^{θ−1} ∈ L1((−∞,0]) is sufficient for b(p0) ≫ 0, and (with Neumann/Robin and 0 ≤ αP ≤ λP) also necessary; its proof hinges on cohomologizing the 1D cocycle to a smooth exponential form, comparing with a scalar logistic-type ODE, and constructing sub/supersolutions along the principal bundle. The candidate solution follows essentially the same route: a 1D cohomology reduction c → \hat c = exp(∫ a), comparison with a time-dependent logistic ODE for amplitudes along the principal eigenbundle, and two-sided bounds via m(s), M(s). The logical endpoints match the paper’s statements. Minor gaps in the candidate solution concern regularity (it asserts a time-derivative identity for e(p·s) without the smoothness provisos that the paper explicitly introduces), but these are technical details already addressed by the paper through the cohomology-to-smooth-cocycle step and classical comparison theorems. Overall, both are correct and essentially the same proof, with the paper providing the needed technical scaffolding (smooth cohomology, comparison principles) and the candidate providing a compatible explicit formula for the scalar ODE bound. Key statements and proof steps appear in Theorem 5.5 and its proof, including the cohomology reduction and the ODE comparison construction, as well as the uniform positivity needed for necessity under Neumann/Robin boundary conditions . Definitions of the principal bundle and cocycle are as in (3.5)–(3.6), and the pullback characterization of b used by both arguments is recalled in (3.3) .