Strongly anisotropic type II blow up at an isolated point

Abstract

We consider the energy supercriticald+1d+1-dimensional semi-linear heat equation∂tu=Δu+up, x∈Rd+1, p≥3, d≥14.\begin{equation*} \partial _tu=\Delta u+u^{p}, \ \ x\in \Bbb R^{d+1}, \ \ p\geq 3, \ d\geq 14. \end{equation*}A fundamental open problem on this canonical nonlinear model is to understand the possible blow-up profiles appearing after renormalisation of a singularity. We exhibit in this paper a new scenario corresponding to the first example of a strongly anisotropic blow-up bubble: the solution displays a completely different behaviour depending on the considered direction in space. A fundamental step of the analysis is to solve thereconnection problemin order to produce finite energy solutions which is the heart of the matter. The corresponding anistropic mechanism is expected to be of fundamental importance in other settings in particular in fluid mechanics. The proof relies on a new functional framework for the construction and stabilisation of type II bubbles in the parabolic setting using energy estimates only, and allows us to exhibit new unexpected blow-up speeds.