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

In this paper, we show the scattering and blow-up result of the radialsolution with the energy below the threshold for the nonlinear Schr\"{o}dingerequation (NLS) with the combined terms iu_t + \Delta u = -|u|^4u + |u|^2u\tag{CNLS} in the energy space $H^1(\R^3)$. The threshold is given by theground state $W$ for the energy-critical NLS: $iu_t + \Delta u = -|u|^4u$. Thisproblem was proposed by Tao, Visan and Zhang in \cite{TaoVZ:NLS:combined}. Themain difficulty is the lack of the scaling invariance. Illuminated by\cite{IbrMN:f:NLKG}, we need give the new radial profile decomposition with thescaling parameter, then apply it into the scattering theory. Our result showsthat the defocusing, $\dot H^1$-subcritical perturbation $|u|^2u$ does notaffect the determination of the threshold of the scattering solution of (CNLS)in the energy space.

The dynamics of the 3D radial NLS with the combined terms