Time behavior of solutions to nonlinear Schrödinger equations

概要

In this paper, we deal with nonlinear Schrödinger system (NLS) in the mass-subcritical case and nonlinear Schrödinger equation with a potential (NLSV ) (or (NLSγ)) in the inter-critical case. We consider time behavior of solutions to these equations. For (NLS), we define a scattering threshold, by focusing structure of the nonlinearity, which corresponds to the best constant of small data scattering. We investigate a property of a solution on the threshold and an optimizing sequence of the threshold. For (NLSV ), we prove a scattering result, a blow-up or grow-up result, and a blow-up result below the ground state without a potential. Then, we show existence of a “radial” ground state and characterize the “radial” ground state by the virial functional. By using the “radial” ground state, we get a global well-posedness of (NLSV ). For (NLSγ), we show blow-up results. Moreover, we obtain equivalence of conditions on initial data below the ground state without a potential by utilizing the global well-posedness results and the blow-up result.