Global existence for the one-dimensional mechanochemical vesicle model

Establish global existence, or characterize the conditions for global well-posedness, for solutions of the one-dimensional Helfrich–reaction–diffusion mechanochemical vesicle model with curvature-dependent morphogen production, spontaneous curvature c_0=\alpha+\beta\phi, and a length constraint.

Background

The paper formulates the one-dimensional vesicle dynamics as a fourth-order geometric evolution equation for a closed planar curve coupled to a reaction–diffusion equation for a morphogen. Local existence follows from the parabolic structure after imposing the length constraint, but the authors do not establish global existence.

Numerical simulations indicate that, for some parameter regimes and initial conditions, solutions may develop self-intersections and apparent blow-up. A rigorous global-existence theory would therefore clarify whether these singular behaviors are genuine properties of the continuum model or artifacts of the numerical discretization, and would identify parameter regimes in which the dynamics remain globally regular.