A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$

Abstract

In this paper we study the threshold of global existenceand blow-up for the solutions to the generalized 3DDavey-Stewartson equations\begin{equation*} \left\{ \begin{aligned} & iu_t + \Delta u + |u|^{p-1} u + E_1(|u|^2)u = 0, \quad t > 0, \ \ x\in \mathbb{R}^3, \\ & u(0,x) = u_0(x) \in H^1(\mathbb{R}^3), \end{aligned} \right.\end{equation*}where $1 < p < \frac{7}{3}$ and the operator $E_1$ is given by$ E_1(f) = \mathcal {F}^{-1} \left( \frac{\xi_1^2}{|\xi|^2} \mathcal{F}(f) \right)$.We construct two kinds of invariant sets underthe evolution flow by analyzing the property of the upper boundfunction of the energy. Then we show that the solution existsglobally for the initial function $u_0$ in first kind of the invariant sets,while the solution blows up in finite time for $u_0$ in another kind.We remark that the exponent $ p $ is subcritical for the nonlinearSchrödinger equations for which blow-up solutions would not occur.The result shows that the occurrence of blow-up phenomenon is caused by thecoupling mechanics of the Davey-Stewartson equations.