Well-posedness and scattering of inhomogeneous cubic-quintic NLS

In this paper, we consider inhomogeneous cubic-quintic nonlinear Schrodinger (ICQNLS) in space dimension d = 3, iu(t) = -Delta u + K-1(x)|u|(2)u + K-2(x)|u|(4)u. We study local well-posedness, finite time blowup, and small data scattering and nonscattering for the ICQNLS when K1,K2 is an element of C4(R3\{0}) satisfy growth condition |partial derivative jKi(x)|less than or similar to|x|bi-j(j=0,1,2,3,4) for some b(i) >= 0 and for x not equal 0. To this end, we use the Sobolev inequality for the functions f is an element of H-n (n = 1, 2) such that |L|lfHn<infinity(l=1,2), where L is the angular momentum operator defined by L = x x (-i del).