Vol. 10, No. 1, 2017

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

Charles Collot

Vol. 10 (2017), No. 1, 127–252
Abstract

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

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

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 ${\left({u}_{\ell }\right)}_{\ell \in ℕ}$ 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.

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Keywords
blow up, heat, soliton, ground state, nonlinear, nonradial, supercritical
Mathematical Subject Classification 2010
Primary: 35B44
Secondary: 35K58, 35B20
Milestones
Received: 9 April 2016
Accepted: 29 September 2016
Published: 30 January 2017
Authors
 Charles Collot Laboratoire J.A. Dieudonné Université de Nice Sophia Antipolis Parc Valrose %, 28 Avenue Valrose 06108 Cedex 02 Nice France