2026/07/18 by Manuel del Pino, Monica Musso, Juncheng Wei +1
#math.AP
We construct a new type II finite-time blow-up mechanism for the energy-supercritical heat equation ut=Δu+u3, n≥ 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 ξr(t)∼ √(2(n-4)(T-t)). At the same time, concentration takes place transversely to the sphere at the much smaller scale λ(t)∼ κ_* \fracT-t|log(T-t)|^\frac nn-2, for some κ_*>0. More precisely, in cylindrical coordinates r=|x'|, z∈\mathbb R3, the leading profile is u(x,t) ∼ (1)/(λ(t)) U( (r-ξr(t))/(λ(t)), (z)/(λ(t)) ), where U is the Aubin--Talenti bubble in \mathbb R4. 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 √(T-t), whereas its transverse thickness is governed by the much smaller type II scale λ(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≥5, but lies below the Joseph--Lundgren exponent for 5≤ n≤ 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.