Dynamics near the ground state for the energy critical nonlinear heat equation in large dimensions

We consider the energy critical semilinear heat equation $${\mathrm{\partial }}_{t}u=\mathrm{\Delta }u+\mathrm{\mid }u{\mathrm{\mid }}^{\frac{4}{d-2}}u,x\in {\mathbb{R}}^d$$ and give a complete classification of the flow near the ground state solitary wave $$Q(x)=\frac{1}{{(1+\frac{\mathrm{\mid }x{\mathrm{\mid }}^2}{d(d-2)})}^{\frac{d-2}{2}}}$$ in dimension $d\ge 7$, in the energy critical topology and without radial symmetry assumption. Given an initial data $Q+{\epsilon }_0$ with $\parallel \nabla \varepsilon_0\parallel_{L^2}\ll 1$, the solution either blows up in the ODE type I regime, or dissipates, and these two open sets are separated by a codimension one set of solutions asymptotically attracted by the solitary wave. In particular, non self similar type II blow up is ruled out in dimension $d\ge 7$ near the solitary wave even though it is known to occur in smaller dimensions. Our proof is based on sole energy estimates deeply and draws a route map for the classification of the flow near the solitary wave in the energy critical setting. A by-product of our method is the classification of minimal elements around $Q$ belonging to the unstable manifold.