Well posedness and smoothing effect of Schrödinger-Poisson equation
In this work we take under consideration the Cauchy problem for the Schrödinger-Poisson type equation i t u=- x2 u+V (u) u-f (∫u∫2) u, where f represents a local nonlinear interaction (we take into account both attractive and repulsive models) and V is taken as a suitable solution of the Poisson equ...
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2007
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00222488_v48_n9_p_DeLeo http://hdl.handle.net/20.500.12110/paper_00222488_v48_n9_p_DeLeo |
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| Sumario: | In this work we take under consideration the Cauchy problem for the Schrödinger-Poisson type equation i t u=- x2 u+V (u) u-f (∫u∫2) u, where f represents a local nonlinear interaction (we take into account both attractive and repulsive models) and V is taken as a suitable solution of the Poisson equation V=12 ∫x∫ (C- ∫u∫2), C Cc∞ is the doping profile or impurities. We show that this problem is locally well posed in the weighted Sobolev spaces Hs { Hs (R): (1+ x2) 12 ∫∫2 <∞} with s1, which means the local existence, uniqueness, and continuity of the solution with respect to the initial data. Moreover, under suitable assumptions on the local interaction, we show the existence of global solutions. Finally, we establish that for s1 local in time and space, smoothing effects are present in the solution; more precisely, in this problem there is locally a gain of half a derivative. © 2007 American Institute of Physics. |
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