Well posedness and smoothing effect of Schroedinger–Poisson equation.

MARIANO DE LEO; DIEGO RIAL

JOURNAL OF MATHEMATICAL PHYSICS

AMER INST PHYSICS

Lugar: New York; Año: 2007 vol. 48 p. 1 - 16

0022-2488

In this work we take under consideration the Cauchy problem forthe Schr¨odinger–Poisson type equation i @tu = −@2xu+V (u) u−f(|u|2) u ,where f represents a local nonlinear interaction (we take into accountboth attractive and repulsive models), V is taken as a suitable solutionof the Poisson equation : V = 1/2 |x| C − |u|2and C 2 C1cis the doping profile, or impurities. We show that this problem is locallywell posed in the weighted R Sobolev spaces Hs := {´ 2 Hs(R) :(1 + x2)1/2 |´|2 < 1} with s 1, which means : local existence,uniqueness and continuity of the solution with respect to the initialdata. Moreover, under suitable assumptions on the local interactionwe show the existence of global solutions. Finally, we establish thatfor s 1 local in time and space smoothing effects are present in thesolution ; more precisely, in this problem there is locally a gain of halfa derivative.