Anal. PDE 10 (1), 127-252, (2017) DOI: 10.2140/apde.2017.10.127
KEYWORDS: blow up, heat, soliton, ground state, nonlinear, nonradial, supercritical, 35B44, 35K58, 35B20

We consider the semilinear heat equation in large dimension $d\ge 11$

$${\partial}_{t}u=\Delta u+|u{|}^{p-1}u,\phantom{\rule{1em}{0ex}}p=2q+1,\phantom{\rule{1em}{0ex}}q\in \mathbb{N},$$

on a smooth bounded domain $\Omega \subset {\mathbb{R}}^{d}$ with Dirichlet boundary condition. In the supercritical range $p\ge p\left(d\right)>1+\frac{4}{d-2}$, we prove the existence of a countable family ${\left({u}_{\ell}\right)}_{\ell \in \mathbb{N}}$ of solutions blowing up at time $T>0$ with type II blow up:

$$\parallel {u}_{\ell}\left(t\right){\parallel}_{{L}^{\infty}}\sim C{\left(T-t\right)}^{-{c}_{\ell}}$$

with blow-up speed ${c}_{\ell}>\frac{1}{p-1}$. The blow up is caused by the concentration of a profile $Q$ which is a radially symmetric stationary solution:

$$u\left(x,t\right)\sim \frac{1}{\lambda {\left(t\right)}^{\frac{2}{p-1}}}Q\left(\frac{x-{x}_{0}}{\lambda \left(t\right)}\right),\phantom{\rule{1em}{0ex}}\lambda \sim C\left({u}_{n}\right){\left(T-t\right)}^{\frac{{c}_{\ell}\left(p-1\right)}{2}}\phantom{\rule{0.3em}{0ex}},$$

at some point ${x}_{0}\in \Omega $. The result generalizes previous works on the existence of type II blow-up solutions, which only existed in the radial setting. The present proof uses robust nonlinear analysis tools instead, based on energy methods and modulation techniques. This is the first nonradial construction of a solution blowing up by concentration of a stationary state in the supercritical regime, and it provides a general strategy to prove similar results for dispersive equations or parabolic systems and to extend it to multiple blow ups.