Collapsing-tube type II blow-up for the energy-supercritical heat equation

Abstract

We construct a new type II finite-time blow-up mechanism for the energy-supercritical heat equation \[ u_t=Δu+u^3, \qquad n\geq 5. \] The solution is positive and blows up only at the origin, but in a highly anisotropic fashion. As $t\nearrow T$, the solution concentrates in a thin tubular region around an $(n-4)$-dimensional sphere whose radius shrinks to zero at the self-similar scale \[ ξ_r(t)\sim \sqrt{2(n-4)(T-t)}. \] At the same time, concentration takes place transversely to the sphere at the much smaller scale \[ λ(t)\sim κ_* \frac{T-t}{|\log(T-t)|^{\frac n{n-2}}}, \] for some ${\kappa }_{\ast }>0$. More precisely, in cylindrical coordinates $r=|x^{\prime }|$, $z\in {\mathbb{R}}^3$, the leading profile is \[ u(x,t) \sim \frac{1}{λ(t)} U\left( \frac{r-ξ_r(t)}{λ(t)}, \frac{z}{λ(t)} \right), \] where $U$ is the Aubin--Talenti bubble in ${\mathbb{R}}^4$. The construction reveals a two-scale singularity mechanism in which a critical transverse bubble concentrates around a geometric set that itself collapses. The concentration tube evolves at the parabolic scale $\sqrt{T-t}$, whereas its transverse thickness is governed by the much smaller type II scale $\lambda (t)$. The logarithmic blow-up law is determined by a nonlocal modulation equation arising from the interaction between the four-dimensional critical bubble and the axisymmetric heat kernel. To our knowledge, this seems to be the first Type II blowup that quantifies the effect of a self-similar collapsing tube. The exponent $p=3$ is energy-supercritical in dimensions $n\ge 5$, but lies below the Joseph--Lundgren exponent for $5\le n\le 12$, in a regime where positive radial type II blow-up is ruled out. The present result provides the first example of a positive type II, single-point blow-up through a collapsing thin-tube geometry.