Threshold and Below-Threshold Dynamics for the Energy-Critical NLS: The Higher-Kinetic-Energy Regime

Abstract

We study the focusing energy-critical nonlinear Schrödinger equation below and at the ground-state energy, focusing on solutions in the upper energy cone, characterized by \[ E(u)\le E(W),\qquad \|\nabla u\|_{L_x^2}>\|\nabla W\|_{L_x^2}, \] where \(W\) is the ground state. Unlike the lower cone, where compactness of critical solutions plays a central role in understanding the dynamics, such compactness is generally unavailable on the upper branch. Consequently, previous results have often required additional assumptions, such as finite mass or finite variance. In this paper, we remove the finite-mass assumption in the radial setting and give a complete characterization of the upper-cone dynamics in the natural energy space \(\dot H^1_{\mathrm{rad}}\), resolving an open problem posed by Duyckaerts and Merle\cite{DM08}. In particular, this establishes finite-time blow-up of the special threshold solution \(W^+\) in the negative time direction in dimensions \(d=3,4\). For non-radial data in dimensions \(d\ge5\), we show that {for solutions in the interior of the upper cone, the limit superior of their $\dot H^1$-norm satisfies an explicit lower bound as time approaches each endpoint of the maximal lifespan,} while forward-global solutions on the upper threshold sheet that remain below this level must coincide with \(W^+\) modulo symmetries. We also obtain a complete classification of upper-threshold dynamics for finite-variance data in all dimensions \(d\ge3\).