Nonlinear angular-thickness instability of collapsing-tube Type-II blow-up

This work studies nonlinear 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 background solution concentrates along an (n−4)-dimensional shrinking sphere, while a four-dimensional critical bubble concentrates transversely at a much smaller logarithmic scale.The analysis builds on the fixed-harmonic angular-thickness spectrum established in the companion linear paper. That spectrum shows that the degree-one thickness sector is relatively unstable with respect to the background transverse scale, whereas every fixed harmonic degree ℓ ≥ 2 is relatively stable. The relative multiplicity of the degree-one angular-thickness sector is n − 3. PAPER_II_FINAL_REVISEDThe main result upgrades this relative linear instability to a nonlinear PDE exit theorem in a source-compatible tuned class.The first new ingredient is an SO(n−3)-equivariant nonlinear normal form for the active degree-one thickness amplitudes. Quadratic self-interaction of degree-one spherical harmonics produces only degree-zero and degree-two components, so there is no direct quadratic forcing back into the active degree-one sector. These quadratic components are solved as slaved fields, and the first nonlinear self-feedback into the active mode occurs at cubic order. PAPER_II_FINAL_REVISEDThis yields a reduced degree-one modulation equation whose nonlinear forcing is cubic up to a controlled quartic remainder. The reduced system is gauge-covariant: the invariant object is the cubic normal-form conjugacy class together with the degree-one memory operator, rather than a coordinate-dependent scalar cubic coefficient. PAPER_II_FINAL_REVISEDA second essential step is the construction of an all-angular inverse in H^s. This cannot be obtained by summing the fixed-harmonic inverses from the linear paper because their constants are allowed to depend on the harmonic degree. Instead, the angular spectrum is split into finitely many low harmonics and a coercive high-frequency block.The low modes are handled by the corrected fixed-harmonic and modulation inverses. The high-frequency inner problem is controlled by retaining the full angular Laplacian in the principal energy. The potentially large spherical-harmonic eigenvalue is absorbed into the principal angular coercivity, while the variable cylindrical coefficient enters only as a relative o(1) error. This produces a uniform all-angular corrected inverse on the tuned source-compatible complement. PAPER_II_FINAL_REVISED PAPER_II_FINAL_REVISEDThe resulting nonlinear inner-outer system is solved by a contraction argument on a pre-exit chart. The complementary inner, outer, modulation, matching, and re-gluing fields remain quadratic or source-small relative to the active degree-one amplitude, and reconstruction gives a true solution of the original semilinear heat equation. PAPER_II_FINAL_REVISEDTo obtain a pointwise relative instability statement, the argument uses a causal Volterra comparison and an invariant profile cone. The nonlinear trajectory remains close to a nonvanishing homogeneous degree-one profile until the normalized angular-thickness amplitude reaches a prescribed small exit threshold. Within this cone, the normalized degree-one amplitude has strictly positive logarithmic growth.Consequently, arbitrarily small source-compatible physical data with nonzero degree-one angular-thickness component generate true nonsymmetric solutions whose normalized thickness perturbation grows to a fixed threshold before the nominal blow-up time. At the exit time, all complementary components remain lower order. PAPER_II_FINAL_REVISEDThe main theorem therefore establishes nonlinear angular-thickness instability of the symmetric collapsing-tube Type-II solution within the constructed source-compatible tuned class. The theorem does not assert post-exit continuation to the blow-up time, the existence of a nonsymmetric Type-II blow-up profile, or that n − 3 is the full unstable index of the unrestricted PDE.