Electronic Journal of Differential Equations, 
Vol. 2021 (2021), No. 24, pp. 1-13.

Title: Small data blow-up of solutions to nonlinear Schrodinger equations without 
      gauge invariance in L^2
       
Authors: Yuanyuan Ren (Dongguan Univ. of Technology, Dongguan, Guangdong, China)
         Yongsheng Li (South China Univ.of Technology, Guangzhou, Guangdong, China)

Abstract:
 In this  article we study the Cauchy problem of the nonlinear Schrodinger equations
 without gauge invariance
 $$\displaylines{ 
 i\partial_{t}u+\Delta u=\lambda(|u|^{p_1}+|v|^{p_2}), \quad
   (t,x)\in[0,T)\times \mathbb{R}^n,\cr
 i\partial_{t}v+\Delta v=\lambda(|u|^{p_2}+|v|^{p_1}), \quad (t,x)\in [0,T)\times \mathbb{R}^n,
 }$$
 where 1&lt;p<sub>1</sub>, p<sub>2</sub>&lt;1+4/n and $\lambda\in \mathbb{C}\backslash\{0\}$.
 We first prove the  existence of a local solution with initial data in 
 L<sup>2</sup>(R<sup>n</sup>).
 Then under a suitable condition on the initial data, we show that the L<sup>2</sup>-norm of the
 solution must blow up in finite time although the initial data are arbitrarily small.
 As a by-product, we also obtain an upper bound of the maximal existence time of the solution.

Submitted February 15, 2018. Published March 31, 2021.
Math Subject Classifications: 35Q55, 35B44.
Key Words: Nonlinear Schrodinger equations; weak solution; blow up of solutions.