Boundary feedback stabilization of a critical nonlinear JMGT equation with Neumann-undissipated part of the boundary

The paper proves Theorems 2.3–2.4 via a two-level linear exponential stability result (Assumption 1 and γ(x)>0), a precise H1-mapping estimate for the nonlinearity F, an invariant small ball in the low norm H, and a Duhamel + Gronwall–Beesack argument leading to exponential decay in H1. Key steps include Lemma 5.1’s quantitative bound τ‖F(Φ)‖_{H1} ≤ C β ‖Φ‖_{H1} for trajectories with sup‖Φ‖_H ≤ β, and the linear semigroup bounds ‖T(t)‖_{L(H1)} ≤ M1 e^{-ω1 t} and ‖S(t)‖_{L(H)} ≤ M0 e^{-ω0 t} (ω1<ω0), yielding Theorem 6.1 and thus Theorem 2.4. By contrast, the candidate solution places F only in H (not H1) and then estimates the H1-solution using ‖T(t−s)F(Φ(s))‖_{H1} ≤ M e^{-ω(t−s)}‖F(Φ(s))‖_{H}, which is not justified since the paper does not establish any smoothing bound T(t): H→H1. Without proving the stronger H1-bound for F or a smoothing property of T(t), the Duhamel estimate in H1 used by the candidate does not close. Hence, the model’s proof has a critical gap, while the paper’s proof is complete.