Some nonexistence results for space-time fractional Schr{\"o}dinger equations without gauge invariance

TL;DR

This paper investigates the nonexistence of solutions for space-time fractional Schrödinger equations in various critical regimes, highlighting conditions under which solutions cannot persist globally or locally.

Contribution

It provides new nonexistence results for semi-linear space-time fractional Schrödinger equations without gauge invariance, covering subcritical, critical, and supercritical cases.

Findings

Nonexistence of global solutions in subcritical and critical cases.

Nonexistence of local solutions in supercritical case.

Conditions on initial data and nonlinear terms for nonexistence.

Abstract

In this paper, we consider the Cauchy problem in $\mathbb{R}^N$, $N\geq1$, for semi-linear Schr\"odinger equations with space-time fractional derivatives. We discuss the nonexistence of global $L^1$ or $L^2$ weak solutions in the subcritical and critical cases under some conditions on the initial data and the nonlinear term. Furthermore, the nonexistence of local $L^1$ or $L^2$ weak solutions in the supercritical case are studied.