Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum

The paper proves sharp t^{-3/2} decay and asymptotics for perturbations of critical pulled fronts under Hypotheses 1–4 using a detailed resolvent expansion in γ = √λ near the essential spectrum and an integration contour tangent to the spectrum (Theorem 2) . It assumes no bounded solution to Lu=0 (no resonance) and no unstable point spectrum (Hypothesis 4), takes an exponential weight ω=ω_{η*} that pushes the right essential spectrum to touch the origin, and identifies a linearly growing neutral solution ψ of Lψ=0 unique up to scaling, exactly as set out in the preliminaries and hypotheses . The candidate solution reproduces these ingredients: the same weighting, the spectral picture with quadratic touching, the absence of a bounded resonance, the rank-one γ term in the resolvent expansion giving the t^{-3/2} law, and the Duhamel/bootstrapping that yields a convergent amplitude α* with remainder O(t^{-2}) in H^1_{−r} for r>5/2—precisely the paper’s Theorem 2 conclusion . Where the candidate uses an Evans/transmission-coefficient viewpoint, the paper uses an explicit Green’s/resolvent construction; these are compatible perspectives, and the logical steps match the paper’s proofs (e.g., (L−γ^2)^{-1}=R_0+γR_1+O(γ^2) with R_1 rank-one, leading to t^{-3/2} via contour integration) . No substantive conflict was found.