Logarithmic Schrödinger equation with quadratic potential*

Abstract We analyze dynamical properties of the logarithmic Schrödinger equation under a quadratic potential. The sign nonlinearity is such that it known in absence external potential, every solution dispersive, with universal asymptotic profile. introduction harmonic potential generates solitary waves, corresponding to generalized Gaussons. prove they are orbitally stable, using an inequality related relative entropy, which may be thought as dual classical Sobolev inequality. In case partial confinement, we show dispersive behavior for suitable marginals. For repulsive potentials, rate dictated by and no must expected.