Analysis & PDE

  • Anal. PDE
  • Volume 10, Number 1 (2017), 127-252.

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

Charles Collot

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Abstract

We consider the semilinear heat equation in large dimension d 11

tu = Δu + |u|p1u,p = 2q + 1,q ,

on a smooth bounded domain Ω d with Dirichlet boundary condition. In the supercritical range p p(d) > 1 + 4 d2, we prove the existence of a countable family (u) of solutions blowing up at time T > 0 with type II blow up:

u(t)L C(T t)c

with blow-up speed c > 1 p1. The blow up is caused by the concentration of a profile Q which is a radially symmetric stationary solution:

u(x,t) 1 λ(t) 2 p1 Q(x x0 λ(t) ),λ C(un)(T t)c(p1) 2 ,

at some point x0 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.

Article information

Source
Anal. PDE, Volume 10, Number 1 (2017), 127-252.

Dates
Received: 9 April 2016
Accepted: 29 September 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843405

Digital Object Identifier
doi:10.2140/apde.2017.10.127

Mathematical Reviews number (MathSciNet)
MR3611015

Zentralblatt MATH identifier
1372.35153

Subjects
Primary: 35B44: Blow-up
Secondary: 35K58: Semilinear parabolic equations 35B20: Perturbations

Keywords
blow up heat soliton ground state nonlinear nonradial supercritical

Citation

Collot, Charles. Nonradial type II blow up for the energy-supercritical semilinear heat equation. Anal. PDE 10 (2017), no. 1, 127--252. doi:10.2140/apde.2017.10.127. https://projecteuclid.org/euclid.apde/1510843405


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