Localized Patterns

The paper (a review) states Theorem 4.3: existence of planar, axisymmetric Spot B solutions with core asymptotics uB(r) ∼ −μ^{3/8} J0(r) + O(√μ) uniformly on [0, r0], under the subcriticality condition |ν| > √(27/38). It sketches the radial spatial-dynamics program (core/far-field decomposition and matching via a nonautonomous Ginzburg–Landau reduction). The candidate solution follows the same program: constructs a core manifold along J0, a 2D far-field stable manifold, performs a blow-up/matching that yields the inner scaling r ∼ a^{-2/3}, μ ∼ a^{8/3}, and the amplitude law a ∼ μ^{3/8}, with the same threshold on ν. The only overreach is the model’s phrasing that existence is guaranteed ‘exactly when’ |ν|>√(27/38); the paper presents this as a sufficient hypothesis used in the proofs, not a proved necessity.