Rigorous stability of stationary pulse solutions

Prove the stability of the stationary pulse solutions constructed for the two-dimensional bulk–surface reaction–diffusion system and establish whether their stability is determined by the signs of the second derivative E''(s_{0,*};w_*) of the geometry-dependent potential and the derivative J'(v_*) of the mass-balance function.

Background

The paper constructs stationary pulse solutions near every nondegenerate critical point s_{0,} of the potential E(·;w_), using a singular perturbation and Lyapunov–Schmidt reduction. The formal pulse-center dynamics derived in earlier work indicate that the pulse motion is governed by the gradient of E, while the sign of J'(v_*) corresponds to the wave-pinning stability condition.

The paper does not analyze the spectral or nonlinear stability of the constructed stationary solutions. It explicitly identifies as unresolved whether the signs of E''(s_{0,};w_) and J'(v_*) rigorously determine pulse stability, leaving this as a problem for future work.

References

The stability of the solutions constructed here has not been investigated in this paper. The ODEs derived formally by the author in suggest that the stability of the pulse is determined by the signs of $E''(s_{0,};w_)$ and $J'(v_*)$. A rigorous proof of this prediction is an interesting question for future work.

— Existence of pulse solutions in a bulk--surface reaction--diffusion system for cell polarization  (2610.11512 - Watanabe, 8 Oct 2026) in Section 6, Conclusion