On blow-up criterion for the nonlinear Schrödinger equation systems

Abstract

In this paper, we study the blow-up problem of nonlinear Schrödinger equations

　　　　　　　　 \{\begin{array}{ll}\text{i}{\mathrm{\partial }}_{t}v+\mathrm{\Delta }u+(|u{|}^2+|v{|}^2)u=0,\hfill & (t,x)\in {\mathbb{R}}^{1+n},\hfill \\ \mathrm{i}{\mathrm{\partial }}_{t}v+\mathrm{\Delta }v+(|u{|}^2+|v{|}^2)v=0,\hfill & (t,x)\in {\mathbb{R}}^{1+n},\hfill \\ u(0,x)=u_0\left(x\right),\hfill & v(0,x)=v_0\left(x\right),\hfill \end{array}

and prove that the solution of negative energy blows up in finite or infinite time.

Key words： Nonlinear Schrödinger equations; blow up; negative energy

References

1 Bergé L. Wave collapse in physics: principles and applications to light and plasma waves. Phys Rep 1998; 303(5/6): 259–370

2 Du D P, Wu Y F, Zhang K J. On blow-up criterion for the nonlinear Schrödinger equation. Discrete Cantin Dyn Syst 2016; 36(7): 3639–3650

3 Glassey R T. On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. J Math Phys 1977; 18(9): 1794–1797

4 Holmer J, Roudenko S. Divergence of infinite-variance nonradial solutions to the 3d NLS equation. Comm Partial Differential Equations 2010; 35(5): 878–905

5 Kenig C, Merle F. Global well-posedness, scattering and blow-up for the energy critical focusing nonlinear wave equation. Acta Math 2008; 201(2): 147–212

6 SulemCSulemP-L. The Nonlinear Schrödinger Equation: Self-focusing and Wave Collapse. New York: Springer-Verlag, 1999