Energy concentration of the focusing energy-critical FNLS

TL;DR

This paper investigates the energy concentration phenomena of radial solutions to the focusing energy-critical fractional nonlinear Schrödinger equation, establishing finite-time blow-up conditions and profile decoupling using advanced analytical techniques.

Contribution

It extends existing methods to the fractional setting, proving energy concentration and finite-time blow-up for solutions near the ground state energy.

Findings

Energy concentrates near the maximal existence time.

Finite-time blow-up occurs when kinetic energy exceeds that of the ground state.

Strong energy decoupling of profiles is established.

Abstract

We consider the fractional nonlinear Schr\"odinger equation (FNLS) with general dispersion $|\nabla|^\alpha$ and focusing energy-critical nonlinearities $-|u|^\frac{2\alpha}{d-\alpha}u$ and $-(|x|^{-2\alpha} * |u|^2) u$. By adopting Kenig-Tsutsumi \cite{mets}, Kenig-Merle \cite{keme} and Killip-Visan \cite{kv} arguments, we show the energy concentration of radial solutions near the maximal existence time. For this purpose we use Sobolev inequalities for radial functions and establish strong energy decoupling of profiles. And we also show that when the kinetic energy is confined the maximal existence time is finite for some large class of initial data satisfying the initial energy $E(\varphi)$ is less than energy of ground state $E(W_\alpha)$ but $\||\nabla|^\frac\alpha2 \varphi\|_{L^2} \ge \||\nabla|^\frac\alpha2 W_\alpha\|_{L^2}$.