Non radial 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 $\Omega \subset {\mathbb{R}}^d$ with Dirichlet boundary condition. In the supercritical range $p\ge p(d)>1+\frac{4}{d-2}$ we prove the existence of a countable family $(u_{\ell }{)}_{\ell \in \mathbb{N}}$ of solutions blowing-up at time $T>0$ with type II blow up: $$\parallel u_{\ell }(t){\parallel }_{L^{\infty }}\sim C(T-t{)}^{-c_{\ell }}$$ with blow-up speed $c_{\ell }>\frac{1}{p-1}$. They concentrate the ground state $Q$ being the only radially and decaying solution of $\Delta Q+Q^p=0$: $$u(x,t)\sim \frac{1}{\lambda (t{)}^{\frac{2}{p-1}}}Q\left(\frac{x-x_0}{\lambda (t)}\right),\lambda \sim C(u_n)(T-t{)}^{\frac{c_{\ell }(p-1)}{2}}$$ 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 non-radial construction of a solution blowing up by concentration of a stationary state in the supercritical regime, and provides a general strategy to prove similar results for dispersive equations or parabolic systems and to extend it to multiple blow ups.

105 pages

References in corpus (1)

• Stable blow up dynamics for the 1-corotational energy critical harmonic heat flow

Cited by in corpus (4)

• Dynamics near the ground state for the energy critical nonlinear heat equation in large dimensions
• On the stability of type II blowup for the 1-corotational energy supercritical harmonic heat flow
• Construction of blow-up manifolds to the equivariant self-dual Chern-Simons-Schrödinger equation
• Existence and stability of infinite time bubble towers in the energy critical heat equation