PUB - Publikationen an der Universität Bielefeld

We review the recent progress on the long-time behavior for a general class of focusing L-2- critical nonlinear Schr & ouml;dinger equations (NLS) with lower order perturbations. Two canonical models are the stochastic NLS driven by linear multiplicative noise and the classical determin-istic NLS. We show the construction and uniqueness of the corresponding blow-up solutions and solitons, including the multi-bubble Bourgain-Wang type blow-up solutions and non-pure multi-solitons, which provide new examples for the mass quantization conjecture and the soliton resolution conjecture. The refined uniqueness of pure multi-bubble blow-ups and pure multi-solitons to NLS under very low asymptotical rate is also reviewed. Finally, as a new result, we prove the qualitative properties of stochastic blow-up solutions, including the concentration of mass, universality of critical mass blow-up profiles, as well as the vanishing of the virial at the blow-up time.