Well-posedness and scattering of inhomogeneous cubic-quintic NLS

Abstract

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

References in corpus (2)

• Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities
• Higher order fractional Leibniz rule