Stability of Asymptotic Waves in the Fisher–Stefan Equation

The paper proves spectral and nonlinear stability for (i) the vanishing state u≡0 on (0,h∞) with ux(t,0)=0, u(t,h∞)=0 via separation of variables, obtaining λn=1−((2n−1)π/(2h∞))2 and the threshold h∞=π/2 for stability, and (ii) the spreading semi-wave ūc on (−∞,0] for 0≤c<2 by computing the essential spectrum using Weyl’s theorem and the far-field dispersion λ=−k2−1+ick, and ruling out point spectrum with Re λ>0 using a Sturm–Prüfer argument; nonlinear stability then follows from Henry’s theory. These steps, equations (5)–(6), (9), (12)–(15), and Theorem 4.2, are explicitly developed in the paper . The candidate solution reproduces (A) the separation-of-variables calculation and threshold exactly, and for (B) gives the same essential spectrum and excludes unstable point spectrum by a ground-state transform using φ(z)=−e^{(c/2)z}ū′c(z)>0, which is an alternative but valid route to the paper’s conclusion. Two small notes: the paper’s brief mention of H^1_0((0,h∞)) for the vanishing problem conflicts with the mixed (Neumann–Dirichlet) boundary conditions, while the candidate’s claim that H^1_0(R_−) is a Banach algebra is not needed (H^1(R_−)→L^4 ensures u↦u^2 maps H^1→L^2). Neither issue affects the main results.