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

We construct a new type II finite-time blow-up mechanism for the energy-supercritical heat equation $$u_t=\mathrm{\Delta }u+u^3,n\ge 5.$$ 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 $${\xi }_r(t)\sim \sqrt{2(n-4)(T-t)}\mathrm{.}$$ At the same time, concentration takes place transversely to the sphere at the much smaller scale $$\lambda (t)\sim {\kappa }_{\ast }\frac{T-t}{\mathrm{\mid }\log (T-t){\mathrm{\mid }}^{\frac{n}{n-2}}},$$ for some ${\kappa }_{\ast }>0$. More precisely, in cylindrical coordinates $r=\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }$, $z\in {\mathbb{R}}^3$, the leading profile is $$u(x,t)\sim \frac{1}{\lambda (t)}U(\frac{r-{\xi }_r(t)}{\lambda (t)},\frac{z}{\lambda (t)}),$$ 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.