Kinetic Behavior Of Generalized DS System

In this paper, we consider the Cauchy problem for the generalized Davey-Stewartson system where a>0,b1b2>0,4/3+1<p<5. By introducing an operator E1f(ξ)=(?)(ξ), the above system can be reduced to the following Cauchy problemWe 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(u0)<J(R), where J stands for the Lagrange functional. By establishing the localized Virial identity, we obtain that when Q(u0)<0, then the solution u blows up both forward and backward in time. The basic strategy to establish the scattering theory is the concentration-compactness arguments from Kenig and Merle [20]. Specifically, suppose that when J(u0)<J(R),Q(u0)>0, the solution u does not scatter. Then by concentration-compactness argument and the blow up result, we will extract the unique bubble and obtain the critical solution. Hence the compactness of the critical element and the localized Virial identity lead to a contrcdiction.