Analysis of Hartree equation with an interaction growing at the spatial infinity

Details Analysis of Hartree equation with an interaction growing at the spatial infinity Analysis of Hartree equation with an interaction growing at the spatial infinity

2011-03-16

arxiv.org/abs/1103.3083v1

Mathematics

Analysis of PDEs

28 pages, no figure

Scientific paper

We consider nonlinear Schr\"odinger equation with Hartree-type nonlinearity. The case where an exponent describing a shape of nonlinearity is negative is studied. In such cases, the nonlinear potential grows at the spatial infinity. Under this situation, we prove the global well-posedness in an energy class. The key for proof is a transformation of the equation by using conservation of mass and conservation of momentum. Because of this respect, uniqueness holds under conservation of momentum. When the nonlinearity grows in the quadratic order, the solution is written explicitly and the uniqueness holds without conservation of momentum. By an explicit representation of the solution, it turns out that this kind of nonlinearity contains an effect like a linear potential.

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