Scattering for the damped inhomogeneous nonlinear Schredinger equation

In this talk, we will present some results on global existence and scattering for the damped inhomogeneous nonlinear Schrödinger equation. We will first discuss a local well-posedness theory which enables us to reach the critical case and to unify results for the homogenous case and the inhomogeneous.
For the non-damped equation we shall discuss some results of global existence for oscillating initial data and scattering theory in a weighted Lebesgue space for a new range. We give a new scattering criterion taking into account the potential. For general potentials, we highlight the impact of the behavior at the origin and infinity on the allowed range. In particular, if the potential is regular, we show that the more it decreases, the more the range giving scattering is wider.
For the damped case, we establish lower and upper bound estimates of the lifespan. We give an explicit dependence on how large the damping is, in terms of the initial data, to ensure the global existence. This fact seems not known in the literature even for the homogeneous case. Also, we prove scattering results with precise decay rates for large damping. This solves an open question raised in the literature for the homogeneous case.
These are joint works with L. Aloui and S.Jbari (University of Tunis El Manar).