Magnetic Virial and applications to blow up for Schr\"odinger and wave equations

TL;DR

This paper establishes conditions under which solutions to certain nonlinear Schrödinger and wave equations with electromagnetic potential blow up in finite time, focusing on negative energy initial states.

Contribution

It provides new blow-up results for focusing mass-critical and supercritical nonlinear Schrödinger equations and energy-subcritical nonlinear wave equations with electromagnetic potentials.

Findings

Blow-up occurs for initial data with negative energy.

Results apply to both Schrödinger and wave equations.

The paper extends blow-up theory to equations with electromagnetic potentials.

Abstract

We prove blow up results for the solution of the initial value problem with negative energy of the focusing mass-critical and supercritical nonlinear Schr\"odinger and the focusing energy-subcritical nonlinear wave equations with electromagnetic potential.