On the Cauchy problem in Sobolev spaces for nonlinear Schrodinger equations with potential

Details On the Cauchy problem in Sobolev spaces for nonlinear Schrodinger equations with potential On the Cauchy problem in Sobolev spaces for nonlinear Schrodinger equations with potential

2006-09-14

arxiv.org/abs/math/0609391v2

Mathematics

Analysis of PDEs

Scientific paper

We consider the Cauchy problem for nonlinear Schrodinger equations in the presence of a smooth, possibly unbounded, potential. No assumption is made on the sign of the potential. If the potential grows at most linearly at infinity, we construct solutions in Sobolev spaces (without weight), locally in time. Under some natural assumptions, we prove that the $H^1$-solutions are global in time. On the other hand, if the potential has a super-linear growth, then the Sobolev regularity of positive order is lost instantly, not matter how large it is, unless the initial datum decays sufficiently fast at infinity.

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