Non-dispersive vanishing and blow up at infinity for the energy critical nonlinear Schr\"odinger equation in R^3

TL;DR

This paper constructs radial solutions to the energy-critical focusing nonlinear Schrödinger equation in three dimensions that asymptotically resemble a rescaled ground state plus radiation, with a specific polynomial scaling law.

Contribution

It demonstrates the existence of solutions with a specific asymptotic form involving a rescaled ground state and radiation, expanding understanding of long-term dynamics in critical NLS.

Findings

Existence of solutions with polynomial scaling behavior

Solutions asymptotically resemble a rescaled ground state plus radiation

Behavior characterized for small scaling exponent 

Abstract

We consider the energy critical focusing NLS in R^3 and prove, for any $\nu$ sufficiently small, the existence of radial finite energy solutions that as $t\to\infty$ behave as a sum of a dynamically rescaled ground state plus a radiation, the scaling law being of the form $t^{\nu}$.