Fixed-harmonic angular-thickness spectrum of collapsing-tube Type-II blow-up

Abstract

This work studies angular-thickness perturbations of the collapsing-tube Type-II blow-up solution for the energy-supercritical semilinear heat equationu_t = Δu + u³, n ≥ 5.The collapsing-tube solution concentrates along an (n−4)-dimensional sphere shrinking at the parabolic scale, while a four-dimensional Aubin–Talenti bubble concentrates transversely at a much smaller logarithmic scale.The paper classifies the corrected angular-thickness linearization in every fixed spherical-harmonic degree ℓ ≥ 1.For a fixed harmonic mode, let−Δ_S Y_(ℓ,k) = ν_ℓ Y_(ℓ,k),withν_ℓ = ℓ(ℓ + n − 5).The main result proves that the corrected fixed-harmonic gluing system is well posed for sufficiently small blow-up time and that every nonzero angular-thickness amplitude satisfiesv_(ℓ,k)(t) = (T − t)^(α_ℓ + o(1)),whereα_ℓ = ℓ(ℓ + n − 5) / [2(n − 4)].In particular, the degree-one sector has exponent α₁ = 1/2 and is relatively unstable with respect to the background transverse concentration scale. By contrast, every fixed degree ℓ ≥ 2 satisfies α_ℓ > 1 and is relatively stable.The relative multiplicity of the degree-one angular-thickness sector is n − 3. This multiplicity refers only to the angular-thickness instability and is not a computation of the full unstable index of the nonlinear PDE. PAPER_I_FINAL_REVISEDThe proof isolates the scale-resonant component of the fixed-harmonic linearized gluing problem and studies the corresponding nonlocal memory operator. The angular heat kernel is described through a shift of the modified-Bessel order. Its early-history, intermediate-time, and near-time regions are analyzed separately.A key point is that the leading logarithmic memory coefficient is independent of the fixed harmonic degree. The degree-dependent Bessel correction contributes only to lower-order terms. Early history gains an additional power, while the microscopic near-time correction is asymptotically negligible.The principal logarithmic coefficient is computed explicitly and shown not to vanish. Consequently, the fixed-harmonic memory operator admits an inverse with logarithmic gain, while the remaining geometric, cutoff, outer, and nonlinear corrections are perturbative. PAPER_I_FINAL_REVISEDAfter coupling the memory, outer, and inner/re-gluing blocks, the corrected fixed-harmonic linear operator is invertible by a Neumann-series argument. This yields the asymptotic modulation equationv̇_(ℓ,k) / v_(ℓ,k)= −ν_ℓ / [2(n − 4)(T − t)] + o((T − t)⁻¹),and hence the stated power law for the angular-thickness amplitude. PAPER_I_FINAL_REVISEDThe theorem is intentionally fixed-harmonic: the constants may depend on ℓ, and no uniform ℓ → ∞ inverse is claimed. The nonlinear all-angular problem requires an additional high-frequency coercive mechanism and is treated separately in the companion work. PAPER_I_FINAL_REVISED추천 필드는: