Blow-up analysis of fast-slow PDEs with loss of hyperbolicity

The paper proves the existence/convergence of slow manifolds for the PDE via a novel PDE blow-up scheme on Galerkin truncations and establishes the transition maps Π_a and Π_e with the µ-dependent dichotomy; the candidate solution obtains the same conclusions using standard NHIM/center-manifold arguments and a 2D reduced ODE blow-up. The approaches differ substantially (paper: PDE blow-up in charts with dynamic domain; model: BLZ + center-manifold + 2D blow-up). The only substantive mismatch is the exit-scale: the paper states k(ε)=O(ε^{1/4}), while the model sketches an O(ε^{1/2}) exit for the reduced ODE and then notes this implies the weaker O(ε^{1/4}) bound; the model does not derive the PDE’s 1/4 exponent from first principles. Overall, both reach the stated claims (T1,T2); the proofs are different, and the model’s argument is less sharp near the singular passage but still consistent with the paper’s bound.