Semilinear Schrödinger Models

Marcelo R. Ebert and Michael Reissig
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Marcelo R. Ebert: University of São Paulo, Department of Computing and Mathematics
Michael Reissig: TU Bergakademie Freiberg, Institute of Applied Analysis

Chapter Chapter 21 in Methods for Partial Differential Equations, 2018, pp 367-382 from Springer

Abstract: Abstract In this chapter we introduce results for semilinear Schrödinger models with power nonlinearity in the focusing and defocusing cases as well. First of all, we show how by a scaling argument a proposal for a critical exponent appears. This critical exponent heavily depends on the regularity of the data. The issue of L 2 and H 1 data is explained. As for the linear Schrödinger equation (see Sect. 11.2.3 ), some conserved quantities are given. Then, a global (in time) well-posedness result is proved for weak solutions in the subcritical L 2 case. This result is valid for both cases focusing and defocusing, respectively. Finally, the subcritical H 1 case is treated. Here the main concern is to show differences between both focusing and defocusing cases. A local (in time) well-posedness result is proved. This result contains, moreover, a blow up result in the focusing and a global (in time) well-posedness result in the defocusing case.

Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-66456-9_21

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DOI: 10.1007/978-3-319-66456-9_21

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