Analysis & PDE
- Anal. PDE
- Volume 10, Number 1 (2017), 127-252.
Nonradial type II blow up for the energy-supercritical semilinear heat equation
We consider the semilinear heat equation in large dimension
on a smooth bounded domain with Dirichlet boundary condition. In the supercritical range , we prove the existence of a countable family of solutions blowing up at time with type II blow up:
with blow-up speed . The blow up is caused by the concentration of a profile which is a radially symmetric stationary solution:
at some point . 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.
Anal. PDE, Volume 10, Number 1 (2017), 127-252.
Received: 9 April 2016
Accepted: 29 September 2016
First available in Project Euclid: 16 November 2017
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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