On Jensen-type inequalities for nonsmooth radial scattering solutions of a loglog energy-supercritical Schrodinger equation

TL;DR

This paper establishes scattering results for radial solutions of a loglog energy-supercritical Schrödinger equation in low regularity spaces, using Jensen-type inequalities to handle the barely supercritical nonlinearity.

Contribution

It introduces Jensen-type inequalities to control nonsmooth solutions of a supercritical Schrödinger equation, enabling scattering analysis in low regularity Sobolev spaces.

Findings

Proves scattering for radial solutions in specified dimensions and regularity.

Develops Jensen-type inequalities for controlling supercritical nonlinearities.

Handles nonsmooth initial data in $ ilde{H}^k$ spaces.

Abstract

Given $n \in \{ 3,4 \}$ (resp. $n=5$) and $k > 1$ (resp. $\frac{4}{3} > k > 1$), we prove scattering of the radial $\tilde{H}^{k}:= \dot{H}^{k}(\mathbb{R}^{n}) \cap \dot{H}^{1}(\mathbb{R}^{n})$ solutions of the loglog energy-supercritical Schrodinger equation $i \partial_{t} u + \triangle u = |u|^{\frac{4}{n-2}}u log^{\gamma} (\log( 10 + |u|^{2} ))$ for $0 < \gamma < \gamma_{n}$. In order to control the barely supercritical nonlinearity for nonsmooth solutions, i.e solutions with data in $\tilde{H}^{k}$, $k \leq \frac{n}{2}$, we prove some Jensen-type inequalities.