Full analysis of high-wavenumber amplitude equations

Investigate and fully analyze the 2ℓ+1-dimensional O(3)-equivariant amplitude equations derived for bulk-surface reaction-diffusion systems posed on a ball with linear bulk kinetics (as in Theorem 1), specifically for larger wavenumbers ℓ, by determining the existence, classification, and stability of their steady and time-dependent solutions.

Background

The paper derives weakly nonlinear amplitude equations capturing Turing bifurcations for bulk-surface reaction-diffusion systems in a 3D ball, exploiting the O(3) symmetry shared with spherical-surface PDEs. For higher wavenumbers ℓ, the resulting amplitude systems have 2ℓ+1 components and become increasingly high-dimensional and intricate.

While the authors compute illustrative examples and provide symmetry-based normal forms, they explicitly note that comprehensive analytical treatment of these large-dimensional amplitude systems is not available. This gap motivates a thorough analysis that would classify branches, bifurcations, and stability across modes and parameter regimes.