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.

Keywords

Nonlinear Schrödinger equations / blow up / negative energy

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Higher Education Press 2023