Non radial type II blow up for the energy supercritical semilinear heat equation

We consider the semilinear heat equation in large dimension $d\ge 11$ $${\mathrm{\partial }}_{t}u=\mathrm{\Delta }u+\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u,p=2q+1,q\in \mathbb{N}$$ on a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$ with Dirichlet boundary condition. In the supercritical range $p\ge p(d)>1+\frac{4}{d-2}$ we prove the existence of a countable family $(u_{\mathrm{\ell }}{)}_{\mathrm{\ell }\in \mathbb{N}}$ of solutions blowing-up at time $T>0$ with type II blow up: $$\parallel u_{\mathrm{\ell }}(t){\parallel }_{L^{\mathrm{\infty }}}\sim C(T-t{)}^{-c_{\mathrm{\ell }}}$$ with blow-up speed $c_{\mathrm{\ell }}>\frac{1}{p-1}$. They concentrate the ground state $Q$ being the only radially and decaying solution of $\mathrm{\Delta }Q+Q^p=0$: $$u(x,t)\sim \frac{1}{\lambda (t{)}^{\frac{2}{p-1}}}Q\left(\frac{x-x_0}{\lambda (t)}\right),\lambda \sim C(u_n)(T-t{)}^{\frac{c_{\mathrm{\ell }}(p-1)}{2}}$$ at some point $x_0\in \Omega$. 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 non-radial construction of a solution blowing up by concentration of a stationary state in the supercritical regime, and provides a general strategy to prove similar results for dispersive equations or parabolic systems and to extend it to multiple blow ups.