Blow up of fractional Schr\"odinger equations on manifolds with nonnegative Ricci curvature

TL;DR

This paper investigates the finite-time blow-up behavior of solutions to fractional Schr"odinger equations with nonlinearities on manifolds with nonnegative Ricci curvature, providing conditions for blow-up and lifespan estimates.

Contribution

It introduces new weight functions and ODE inequalities to analyze blow-up phenomena for fractional Schr"odinger equations on curved manifolds.

Findings

Solutions blow up in finite time under certain volume conditions.

The lifespan of solutions can be explicitly estimated.

Blow-up occurs regardless of initial data size.

Abstract

In this paper, the well-posedness of Cauchy's problem of fractional Schr\"odinger equations with a power type nonlinearity on $n$-dimensional manifolds with nonnegative Ricci curvature is studied. Under suitable volume conditions, the local solution with initial data in $H^{[\frac{n}{2}]+1}$ will blow up in finite time no matter how small the initial data is, which follows from a new weight function and ODE inequalities. Moreover, the upper-bound of the lifespan can be estimated.