2002/10/31 by Remi Carles
Mathematics · Physics and Astronomy · #math.AP #math-ph #math.MP #msc:35Q55 #msc:35B05 #msc:35B40 #msc:35P25
published as SIAM J. Math. Anal., 35, #4 (2003), 823-843. · Final version, to appear in SIAM J. Math. Anal
arxiv created 2003/03/14 · arxiv updated 2009/11/30
We study the Cauchy problem for Schrodinger equations with repulsive quadratic potential and power-like 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.