NONRADIAL TYPE II BLOW UP FOR THE ENERGY-SUPERCRITICAL SEMILINEAR HEAT EQUATION

We consider the semilinear heat equation in large dimension d ≥ 11 [inline-equation] on a smooth bounded domain Ω ⊂ ℝd with Dirichlet boundary condition. In the supercritical range [inline-equation], we prove the existence of a countable family (uℓ)ℓ∈ of solutions blowing up at time T > 0 with type II blow up: [inline-equation] with blow-up speed [inline-equation]. The blow up is caused by the concentration of a profile Q which is a radially symmetric stationary solution: [inline-equation] 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. © 2017 Mathematical Sciences Publishers.