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

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

∥ u_{ℓ} (t) ∥_{L^{\infty}} \sim C {(T - t)}^{- c_{ℓ}}

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

u (x, t) \sim \frac{1}{λ {(t)}^{\frac{2}{p - 1}}} Q (\frac{x - x_{0}}{λ (t)}), λ \sim C (u_{n}) {(T - t)}^{\frac{c_{ℓ} (p - 1)}{2}},

at some point $x_{0} \in Ω$ . 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.

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

Vol. 10, No. 1, 2017

Charles Collot

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors