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Nonlinear Schrödinger equations with repulsive harmonic potential and applications. (English) Zbl 1054.35090

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.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
35P25 Scattering theory for PDEs
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