On the Cauchy problem of the nonlinear Schrödinger equation without gauge invariance

ABSTRACT

This paper is dedicated to study the Cauchy problem of the nonlinear Schrödinger equation without gauge invariance $$\begin{array}{c}\hfill \left\{\begin{array}{cc}i{u}_t+\mathrm{\Delta }u={\lambda (|u|}^{p_1}+{|v|}^{p_2}),\hfill & (t,x)\in [0,T)\times {\mathbb{R}}^n,\hfill \\ i{v}_t+\mathrm{\Delta }v={\lambda (|u|}^{p_2}+{|v|}^{p_1}),\hfill & (t,x)\in [0,T)\times {\mathbb{R}}^n,\hfill \end{array}\right.\end{array}$$

where $p_1,p_2>1$ and $\lambda \in \mathbb{C}\backslash \{0\}$. When $1<p_1,p_2<1+\frac{4}{n-2}$, we first prove local well-posedness of the equation in $H^1({\mathbb{R}}^n)$. If in addition, $1+\frac{4}{n}\le p_1,p_2<1+\frac{4}{n-2}$, we prove the global well-posedness with small initial data in $H^1({\mathbb{R}}^n)$. Under a suitable condition on the initial data and $min\{p_1,p_2\}<1+\frac{4}{n}$, we prove that the $H^1$-norm of the solution would blow up in finite time although the initial data are arbitrarily small. Meanwhile, we also give a large initial data blow-up result when $p_1,p_2<1+\frac{4}{n-2}$ in $H^1({\mathbb{R}}^n)$. Finally, we show the non-existence of local weak solution for some $H^m$-data with $m=0,1$ when $max\{p_1,p_2\}>1+\frac{4}{n-2m}$.

KEYWORDS:

• Nonlinear Schrödinger equation
• weak solution
• non-gauge invariance
• blow up

AMS SUBJECT CLASSIFICATIONS:

• 35Q55
• 35B44

Acknowledgements

The authors would like to express their great gratitude to the referees for their valuable suggestions, which lead to improvements of the paper. Especially, the authors gratefully acknowledge the many helpful suggestions of Meiling Yang during the revision of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by NSFC [grant number 11571118], [grant number 11771127], [grant number 11401180]; and by the Fundamental Research Funds for the Central Universities of China [grant number 2017ZD094].