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

Abstract

We consider the energy critical semilinear heat equation $${\partial }_{t}u=\Delta u+|u{|}^{\frac{4}{d-2}}u,x\in {\mathbb{R}}^d$$ 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 $Q+{\epsilon }_0$ with $\parallel \nabla {\epsilon }_0{\parallel }_{L^2}\ll 1$, 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.

81 pages

References in corpus (2)

• Stable blow up dynamics for the 1-corotational energy critical harmonic heat flow
• Non radial type II blow up for the energy supercritical semilinear heat equation

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