Well-posedness and scattering of inhomogeneous cubic-quintic NLS

In this paper we consider inhomogeneous cubic-quintic NLS in space dimension $d=3$: $$i{u}_t=-\mathrm{\Delta }u+K_1(x)\mathrm{\mid }u{\mathrm{\mid }}^{2}u+K_2(x)\mathrm{\mid }u{\mathrm{\mid }}^{4}u\mathrm{.}$$ We study local well-posedness, finite time blowup, and small data scattering and non-scattering for the ICQNLS when $K_1, K_2 \in C^4(\mathbb R^3 \setminus {0})$satisfygrowthcondition$|\partial^j K_i(x)| \lesssim |x|^{b_i-j}, (j = 0, 1, 2, 3, 4)$forsome$b_i \ge 0$andfor$x \neq 0$. To this end we use the Sobolev inequality for the functions $f\in H^n(n=1,2)$ such that $\mathrm{\parallel }\mathrm{\mid }\mathbf{L}{\mathrm{\mid }}^{\mathrm{\ell }}f{\mathrm{\parallel }}_{H^n}<\mathrm{\infty }(\mathrm{\ell }=1,2)$, where $\mathbf{L}$ is the angular momentum operator defined by $\mathbf{L}=x\times (-i\mathrm{\nabla })$.