Global solutions for $H^s$-critical nonlinear biharmonic Schr\"{o}dinger equation

TL;DR

This paper establishes the existence and uniqueness of global solutions for the critical nonlinear biharmonic Schrödinger equation in Sobolev spaces, focusing on small initial data in various dimensions.

Contribution

It provides the first rigorous proof of global well-posedness for the $H^s$-critical BNLS with small initial data across different dimensions.

Findings

Global solutions exist for small initial data

Uniqueness of solutions in the critical Sobolev space

Applicable for various dimensions and parameters

Abstract

We consider the nonlinear biharmonic Schr\"odinger equation $$i\partial_tu+(\Delta^2+\mu\Delta)u+f(u)=0,\qquad (\text{BNLS})$$ in the critical Sobolev space $H^s(\R^N)$, where $N\ge1$, $\mu=0$ or $-1$, $0<s<\min\{\fc N2,8\}$ and $f(u)$ is a nonlinear function that behaves like $\lambda\left|u\right|^{\alpha}u$ with $\lambda\in\mathbb{C},\alpha=\frac{8}{N-2s}$. We prove the existence and uniqueness of the global solutions to (BNLS) for the small initial data.