Journal
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
Volume 35, Issue 4, Pages 823-843Publisher
SIAM PUBLICATIONS
DOI: 10.1137/S0036141002416936
Keywords
nonlinear Schrodinger equation; finite-time blow-up; scattering
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We study the Cauchy problem for Schrodinger equations with repulsive quadratic potential and powerlike nonlinearity. The local problem is well-posed in the same space as that used when a con. ning 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.
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