Global solutions to 3D quadratic nonlinear Schr\"odinger-type equation
Analysis of PDEs
2026-01-14 v2
Abstract
We consider the Cauchy problem to the 3D fractional Schr\"odinger equation with quadratic interaction of type. We prove the global existence of solutions and scattering properties for small initial data. For the proof, one novelty is that we combine the normal form methods and the space-time resonance methods. Using the normal form transform enables us more flexibilities in designing the resolution spaces so that we can control various interactions. It is also convenient for the final data problem.
Keywords
Cite
@article{arxiv.2409.10804,
title = {Global solutions to 3D quadratic nonlinear Schr\"odinger-type equation},
author = {Zihua Guo and Naijia Liu and Liang Song},
journal= {arXiv preprint arXiv:2409.10804},
year = {2026}
}