Nonexistence of non-trivial global weak solutions for higher-order nonlinear Schrodinger equations

We study the initial-value problem for the higher-order nonlinear Schrodinger equation $$ i\partial_{t}u-(-\Delta)^{m}u=\lambda| u|^{p}, $$ subject to the initial data $$ u(x,0)=f(x), $$ where $u=u(x,t)\in\mathbb{C}$ is a complex-valued function, $(x,t)\in\mathbb{R}^{N}\times[0,+\infty)$, $p>1$, $m\geq 1$, $\lambda\in\mathbb{C}\backslash\{0\},$ and $f(x)$ is a given complex-valued function. We prove nonexistence of a nontrivial global weak solution. Furthermore, we prove that the $L^2$-norm of the local in time $L^2$-solution blows up at a finite time