Geometric blow-up of a dynamic Turing instability in the Swift-Hohenberg equation

The paper’s Theorem 3.20 and its proof are coherent and technically justified via a blow-up framework, evolution-family (semigroup) bounds in uniformly local Sobolev spaces, and sharp residual estimates. By contrast, the model’s outline relies on an unproved time-dependent spectral splitting P_c(t) ⊕ P_s(t) with a uniform gap ≃ r(t)^2 in H^θ_ul, misattributes these bounds to Lemma 3.16, and assumes modulation orthogonality that would require controlling d/dt of the moving projector. The final estimates it claims match the paper’s results, but the key steps it uses are not rigorously justified in the stated function spaces, and some details (e.g., residual in H^{θ−4}_ul) depart from the paper’s technical setup.