Existence and Stability Theory of a Neurologically-Inspired Parabolic PDE Model with a Nonlinear Time-Delayed Boundary Condition

The paper derives the same spectral conditions and conclusions as the candidate solution: for Q<1 all eigenvalues satisfy Re λ<0 for any τ>0, and for Q>1 there is a unique τ0 in (π/(2√(Q^4−1)), π/√(Q^4−1)) where the only imaginary eigenvalues are ± i√(Q^4−1), with positive crossing speed. The candidate solution gives a cleaner, more direct argument using the modulus identity |ρ| = Q e^{-τ Re λ} and an explicit formula for dλ/dτ, while the paper sketches these results via real–imaginary decomposition and an implicit function approach. Minor presentational issues in the paper (e.g., a sketchy uniqueness argument in Theorem 5 and a cumbersome derivative computation) do not change the correctness of the main claims. Overall, both are correct, with different proof styles. Key steps and formulas match the paper’s setup and results (e.g., eigenvalue problem (47)–(52), characteristic equations (54)–(55), relations (56)–(57), (61)–(62), and Theorem 5) .