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Sharp threshold of global existence for nonlinear Schrödinger equation with partial confinement. (English) Zbl 1437.35644
Summary: This paper deals with the nonlinear Schrödinger equation with partial confinement, which may model the attractive Bose-Einstein condensate under a partial trap potential. By using the variational characteristic of the classic nonlinear scalar field equation and the Hamilton conservation, we get the threshold for global existence of the Cauchy problem on mass. Moreover by a novel scaling argument, we prove that this threshold is sharp. In addition, by a numerical computation, we obtain the numerical result of the threshold.
##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35B33 Critical exponents in context of PDEs 35B35 Stability in context of PDEs 82C10 Quantum dynamics and nonequilibrium statistical mechanics (general) 35A01 Existence problems for PDEs: global existence, local existence, non-existence
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