On the focusing mass critical problem in six dimensions with splitting spherically symmetric initial data

摘要

In this paper, we consider the six-dimensional focusing mass critical NLS: iu(t) + Delta u = -vertical bar u vertical bar(2/3) u with splitting-spherical initial data u(0)(x(1), ... x(6)) = u(0)(root x(1)(2) + x(2)(2) + x(3)(2), root x(4)(2) + x(5)(2) + x(6)(2)). we prove that any finite mass solution which is almost periodic modulo scaling in both time directions must have Sobolev regularity H-x(1+). Moreover, the kinetic energy of the solution is localized around the spatial origin uniformly in time. As important applications of the results, we prove the scattering conjecture for solutions with mass smaller than that of the ground state. We also prove that any two-way non-scattering solution must be global and coincides with the solitary wave up to symmetries. Here the ground state is the unique positive, radial solution of the nonlinear elliptic equation Delta Q - Q + Q(5/3) = 0. To prove the smoothness of the solution, we use a new local iteration scheme which first appears in [19].