Adami R., Teta A.
A Class of Nonlinear Schroedinger Equations with Concentrated Nonlinearity
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ABSTRACT. We consider the nonlinear Schr\"{o}dinger equation in dimension one
with a nonlinearity concentrated in a finite number of points.
Detailed results on the local existence of the solution in fractional
Sobolev spaces
$H^{\rho}$ are given.
We also prove the conservation of the $L^{2}$-norm and the energy
of the solution and give a global existence result for repulsive and
weakly attractive interaction in the space
$H^{1}$. Finally we prove the existence of blow-up solutions for
strongly attractive interaction.