Carles, Rémi Nonlinear Schrödinger equations with repulsive harmonic potential and applications. (English) Zbl 1054.35090 SIAM J. Math. Anal. 35, No. 4, 823-843 (2003). Summary: We study the Cauchy problem for Schrödinger equations with repulsive quadratic potential and powerlike nonlinearity. The local problem is well-posed in the same space as that used when a confining harmonic potential is involved. For a defocusing nonlinearity, it is globally well-posed, and a scattering theory is available, with no long range effect for any superlinear nonlinearity. When the nonlinearity is focusing, we prove that choosing the harmonic potential sufficiently strong prevents blow-up in finite time. Thanks to quadratic potentials, we provide a method to anticipate, delay, or prevent wave collapse; this mechanism is explicit for critical nonlinearity. Cited in 2 ReviewsCited in 37 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 35B40 Asymptotic behavior of solutions to PDEs 35P25 Scattering theory for PDEs Keywords:nonlinear Schrödinger equation; finite-time blow-up; scattering; nonlinearity; Cauchy problem PDFBibTeX XMLCite \textit{R. Carles}, SIAM J. Math. Anal. 35, No. 4, 823--843 (2003; Zbl 1054.35090) Full Text: DOI arXiv