Nonradial type II blow up for the energy-supercritical semilinear heat equation

Abstract

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

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

on a smooth bounded domain $\Omega \subset {ℝ}^d$ 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:

$$u\left(x,t\right)\sim \frac{1}{\lambda {\left(t\right)}^{\frac{2}{p-1}}}Q\left(\frac{x-x_0}{\lambda \left(t\right)}\right),\lambda \sim C\left(u_n\right){\left(T-t\right)}^{\frac{c_{\ell }\left(p-1\right)}{2}},$$

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.