Changing blow-up time in nonlinear Schrödinger equations

Solutions to nonlinear Schrödinger equations may blow up in finite time. We study the influence of the introduction of a potential on this phenomenon. For a linear potential (Stark effect), the blow-up time remains unchanged, but the location of the collapse is altered. The main part of our study concerns isotropic quadratic potentials. We show that the usual (confining) harmonic potential may anticipate the blow-up time, and always does when the power of the nonlinearity is L 2 -critical. On the other hand, introducing a “repulsive” harmonic potential prevents finite time blow-up, provided that this potential is sufficiently “strong”. For the L 2 -critical nonlinearity, this mechanism is explicit : according to the strength of the potential, blow-up is first delayed, then prevented.