Characterising exchange of stability in scalar reaction-diffusion equations via geometric blow-up

The paper’s Theorems 3.2 and 3.3 state exactly the exchange-of-stability estimates at time T=2ρ/ε in the Z̃ norm with errors O(e^{-γρ^2/(2ε)}), with branch selection in the pitchfork case determined by sign(λ) (see Theorem 3.2 and Theorem 3.3; also the concluding assembly in Section 4.2.4) . The model’s solution reaches the same conclusions but by a different route: order-preserving comparison with constant-in-space ODE sub/super-solutions, exponential damping of spatial derivatives in the weighted Z̃-norm, and ODE slow-passage estimates. The paper proves tracking in the rescaling chart via a Grönwall bound on an error equation using the heat semigroup bound ∥e^{tΔ}∥_{Z̃→Z̃}≤1 (Appendix A.4) and then composes chart-to-chart transition maps to obtain the exponentially small corrections after blow-down . Minor gaps in the model (e.g., a slightly optimistic bound a1≤−c|μ|+Cρ^2 that can be tightened to a1≤−c|μ|+Cρ, and an explicit well-posedness/maximum-principle assumption on ℝ) are readily fixable. Net: both arguments are consistent and yield the same quantitative estimates; the paper’s proof is via geometric blow-up and center manifolds, the model’s via comparison and damping.