Global well-posedness, scattering and blow-up for the energy-critical, Schr\"odinger equation with general nonlinearity in the radial case

TL;DR

This paper establishes well-posedness and scattering results for the energy-critical Schrödinger equation with general nonlinearities in the radial case, extending previous findings to broader nonlinear growth conditions.

Contribution

It generalizes existing results by proving well-posedness and scattering for a wider class of nonlinearities satisfying Sobolev critical growth.

Findings

Proved well-posedness in appropriate function spaces.

Established scattering asymptotics for solutions.

Extended previous results to general nonlinearities.

Abstract

In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schr\"odinger equation with general nonlinearity \begin{equation*} \left\{\begin{array}{l} i \partial_t u+\Delta u + f(u)=0,\ (x, t) \in \mathbb{R}^N \times \mathbb{R}, \\ \left.u\right|_{t=0}=u_0 \in H ^1(\mathbb{R}^N), \end{array}\right. \end{equation*} where $f:\mathbb{C}\rightarrow \mathbb{C}$ satisfies Sobolev critical growth condition. Using contraction mapping method and concentration compactness argument, we obtain the well-posedness theory in proper function spaces and scattering asymptotics. This paper generalizes the conclusions in \cite{KCEMF2006}(Invent. Math).