In this talk we consider inhomogeneous cubic-quintic NLS in space dimension $$extract_itex$$d=3$$/extract_itex$$ $$extract_itex$$d = 3$$/extract_itex$$ :
$$extract_itex$$rm\left(ICQNLS\right)quadi{u}_{t}=-Deltau+{K}_{1}\left(x\right)|u{|}^{2}u+{K}_{2}\left(x\right)|u{|}^{4}u.$$/extract_itex$$
We discuss local well-posedness, finite time blowup, and small data scattering and non-scattering for the ICQNLS when $$extract_itex$${K}_{1},{K}_{2}inC\left(mathbb{R}^{3}\right)cap{C}^{4}\left(mathbb{R}^{3}setminus0\right)$$/extract_itex$$ $$extract_itex$$K_1, K_2 in C(mathbb R^3) cap C^4(mathbb R^3 setminus {0})$$/extract_itex$$ satisfy growth condition $$extract_itex$$|partia{l}^{j}{K}_{i}\left(x\right)|lesssim|x{|}^{{b}_{i}-j}\left(j=0,1,2,3\right)$$/extract_itex$$ $$extract_itex$$|partial^j K_i(x)| lesssim |x|^{b_i-j} (j = 0, 1, 2, 3)$$/extract_itex$$ for some $$extract_itex$${b}_{i}>0$$/extract_itex$$ $$extract_itex$$b_i > 0$$/extract_itex$$ and for $$extract_itex$$xneq0$$/extract_itex$$ $$extract_itex$$x neq 0$$/extract_itex$$ . To this end we use the Sobolev inequality for the functions $$extract_itex$$fin{H}^{1}$$/extract_itex$$ $$extract_itex$$f in H^1$$/extract_itex$$ such that $$extract_itex$$|mathbfLf{|}_{{H}^{1}} $$extract_itex$$|mathbf L f|_{H^1} < infty$$/extract_itex$$ , where $$extract_itex$$mathbfL$$/extract_itex$$ $$extract_itex$$mathbf L$$/extract_itex$$ is the angular momentum operator defined by $$extract_itex$$mathbfL=xtimes\left(-inabla\right)$$/extract_itex$$ $$extract_itex$$mathbf L = x times (-inabla)$$/extract_itex$$ .