  We study the Cauchy problem for a kinetic equation arising in the weak
turbulence theory for the cubic nonlinear Schr\"odinger equation. We define
suitable concepts of weak and mild solutions and prove local and global well
posedness results. Several qualitative properties of the solutions, including
long time asymptotics, blow up results and condensation in finite time are
obtained. We also prove the existence of a family of solutions that exhibit
pulsating behavior.
