Vol. 10, No. 1, 2017

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

Charles Collot

Vol. 10 (2017), No. 1, 127–252

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.

blow up, heat, soliton, ground state, nonlinear, nonradial, supercritical
Mathematical Subject Classification 2010
Primary: 35B44
Secondary: 35K58, 35B20
Received: 9 April 2016
Accepted: 29 September 2016
Published: 30 January 2017
Charles Collot
Laboratoire J.A. Dieudonné
Université de Nice Sophia Antipolis
Parc Valrose
%, 28 Avenue Valrose
06108 Cedex 02 Nice