English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We consider the energy critical semilinear heat equation $$\partial_tu=\Deltau+|u|^{\frac{4}{d-2}}u, \ \ x\in \mathbb R^d$$ and give a completeclassification of the flow near the ground state solitary wave$$Q(x)=\frac{1}{\left( 1+\frac{|x|^2}{d(d-2)}\right)^{\frac{d-2}{2}}}$$ indimension $d\ge 7$, in the energy critical topology and without radial symmetryassumption. Given an initial data $Q+\varepsilon_0$ with $\parallel \nabla\varepsilon_0\parallel_{L^2}\ll 1$, the solution either blows up in the ODEtype I regime, or dissipates, and these two open sets are separated by acodimension one set of solutions asymptotically attracted by the solitary wave.In particular, non self similar type II blow up is ruled out in dimension $d\ge7$ near the solitary wave even though it is known to occur in smallerdimensions. Our proof is based on sole energy estimates deeply and draws aroute map for the classification of the flow near the solitary wave in theenergy critical setting. A by-product of our method is the classification ofminimal elements around $Q$ belonging to the unstable manifold.

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