In this talk we consider inhomogeneous cubic-quintic NLS in space dimension
d=3
:
rm(ICQNLS)quadiu t =−Deltau+K 1 (x)|u| 2 u+K 2 (x)|u| 4 u.
We discuss local well-posedness, finite time blowup, and small data scattering and non-scattering for the ICQNLS when K 1 ,K 2 inC(mathbbR 3 )capC 4 (mathbbR 3 setminus0)
satisfy growth condition |partial j K i (x)|lesssim|x| b i −j (j=0,1,2,3)
for some b i >0
and for xneq0
. To this end we use the Sobolev inequality for the functions finH 1
such that |mathbfLf| H 1 <infty
, where mathbfL
is the angular momentum operator defined by mathbfL=xtimes(−inabla)
.
