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

• 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) Mathematics Subject Classification: Primary 35Q53; Secondary 47J35.  Citation: •  [1] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175. [2] T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Métedos Matemáticos, Vol. 22, I.M.U.F.R.J., Rio de Janiero, 1989. [3] R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system, Comm. Part. Diff. Eq., 17 (1992), 967-988.doi: 10.1080/03605309208820872. [4] R. 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