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 {i partial derivative(t)u + Delta u = -a vertical bar u vertical bar(p-1)u + b(1)uv(x1), (t, x) is an element of R x R-3 -Delta v = b(2)(vertical bar u vertical bar(2))(x1), where a > 0,b(1)b(2) > 0, 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 [IT]. We overcome the main difficulties coming from the lack of scaling invariance and the asymmetrical structure of non-linearity (in particular, the nonlinearity is non-local). Furthermore, we adapt the standard method from [9] to obtain the blow up criterion.