Sharp threshold for scattering of a generalized Davey-Stewartson system in three dimension

Abstract

In this paper, we consider the Cauchy problem for the generalized Davey-Stewartson system \begin{eqnarray} &i\partial_t u + \Delta u =-a|u|^{p-1}u+b_1uv_{x_1}, (t,x)\in R \times R^3,\\ &-\Delta v=b_2(|u|^2)_{x_1}, \end{eqnarray} where $a>0,b_1b_2>0$, $\frac{4}{3}+1< p< 5$. We first use a variational approach to give a dichotomy of blow-up and scattering for the solution of mass supercritical equation with the initial data satisfying $J(u_0)<J(R)$, where $J$ stands for the Lagrange functional. The basic strategy is the concentration-compactness arguments from Kenig and Merle [17]. We overcome the main difficulties coming from the lack of scaling invariance and the asymmetrical structure of nonlinearity (in particular, the nonlinearity is non-local). Furthermore, we adapt the standard method from [9] to obtain the blow up criterion.