## 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

#### Abstract

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

$∂tu = Δu + |u|p−1u,p = 2q + 1,q ∈ ℕ,$

on a smooth bounded domain $Ω ⊂ ℝd$ with Dirichlet boundary condition. In the supercritical range $p ≥ p(d) > 1 + 4 d−2$, we prove the existence of a countable family $(uℓ)ℓ∈ℕ$ of solutions blowing up at time $T > 0$ with type II blow up:

$∥uℓ(t)∥L∞ ∼ C(T − t)−cℓ$

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

$u(x,t) ∼ 1 λ(t) 2 p−1 Q(x − x0 λ(t) ),λ ∼ C(un)(T − t)cℓ(p−1) 2 ,$

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
Accepted: 29 September 2016
First available in Project Euclid: 16 November 2017

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

#### 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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