A GLOBAL EXISTENCE AND BLOW-UP THRESHOLD FOR DAVEY-STEWARTSON EQUATIONS IN R3

In this paper we study the threshold of global existence and blowup for the solutions to the generalized 3D Davey-Stewartson equations { iu(t) + Delta u + vertical bar u vertical bar(p-1)u + E-1 (vertical bar u vertical bar(2))u = 0, t > 0, x is an element of R-3 { u(0, ) = u(0)(x) is an element of H-1 (R-3) where 1 < p < 7/3 and the operator E-1 is given by E-1(f) = F-1 (xi(2)(1)/vertical bar xi vertical bar(2) F(f)). We construct two kinds of invariant sets under the evolution flow by analyzing the property of the upper bound function of the energy. Then we show that the solution exists globally 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 nonlinear Schrodinger equations for which blow-up solutions would not occur. The result shows that the occurrence of blow-up phenomenon is caused by the coupling mechanics of the Davey-Stewartson equations.