Blow-up results for systems of nonlinear Schr\"odinger equations with quadratic interaction

TL;DR

This paper proves finite-time blow-up and grow-up phenomena for systems of nonlinear Schrödinger equations with quadratic interactions, extending previous results and providing new bounds in anisotropic and radial cases.

Contribution

It introduces new blow-up and grow-up results for quadratic NLS systems in anisotropic spaces, including polynomial lower bounds on kinetic energy for global solutions.

Findings

Finite-time blow-up for cylindrical symmetric solutions.

Polynomial lower bounds on kinetic energy in the mass-critical case.

Extension of results to general NLS systems with quadratic interactions.

Abstract

We establish blow-up results for systems of NLS equations with quadratic interaction in anisotropic spaces. We precisely show finite time blow-up or grow-up for cylindrical symmetric solutions. With our construction, we moreover prove some polynomial lower bounds on the kinetic energy of global solutions in the mass-critical case, which in turn implies grow-up along any diverging time sequence. Our analysis extends to general NLS systems with quadratic interactions, and it also provides improvements of known results in the radial case.