BLOWING-UP TIME-DEPENDENT SOLUTIONS OF THE PLANAR, CHERN-SIMONS GAUGED NONLINEAR SCHRODINGER-EQUATION

Time-dependent solutions of the two-dimensional Chem-Simons gauged non linear Schrodinger equation are investigated in terms of an initial-va lue problem. We prove that this Cauchy problem is locally well posed i n H-2(R(2)), and that global solutions exist in H-1(R(2)) provided tha t the initial data are small enough in L(2)(R(2)). On the other hand, under certain conditions ensuring, for example a negative Hamiltonian, solutions blow up in a finite time which only depends on the initial data. The diverging shape of collapsing structures is finally discusse d throughout a self-similar analysis.