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

TL;DR

This paper classifies the behavior of solutions near the ground state for the energy critical nonlinear heat equation in large dimensions, showing solutions either blow up or dissipate, with no intermediate blow-up types in dimensions d≥7.

Contribution

It provides a complete classification of flow near the ground state for the energy critical nonlinear heat equation in dimensions d≥7 without radial symmetry.

Findings

Solutions either blow up or dissipate near the ground state.

Non self similar type II blow up is ruled out in dimensions d≥7.

Solutions attracted to the solitary wave form a codimension one set.

Abstract

We consider the energy critical semilinear heat equation $$\partial_tu=\Delta u+|u|^{\frac{4}{d-2}}u, \ \ x\in \mathbb R^d$$ and give a complete classification of the flow near the ground state solitary wave $$Q(x)=\frac{1}{\left( 1+\frac{|x|^2}{d(d-2)}\right)^{\frac{d-2}{2}}}$$ in dimension $d\ge 7$, in the energy critical topology and without radial symmetry assumption. Given an initial data $Q+\varepsilon_0$ 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.