Global existence on nonlinear Schrödinger-IMBq equations

Abstract

In this paper, we consider the Cauchy problem of Schrödinger-IMBq equations in $\mathbb{R}^{n}$, $n \geq 1$. We first show the global existence and blowup criterion of solutions in the energy space for the 3 and 4 dimensional system without power nonlinearity under suitable smallness assumption. Secondly the global existence is established to the system with $p$-powered nonlinearity in $H^{s}(\mathbb{R}^{n})$, $n = 1,2$ for some $\frac{n}{2} < s < \mathrm{min}(2, p)$ and some $p > \frac{n}{2}$ . We also provide a blowup criterion for $n = 3$ in Triebel-Lizorkin space containing BMO space naturally.