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.

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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

Vol. 10, No. 1, 2017

Charles Collot

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors