# Well-posedness and scattering of inhomogeneous cubic-quintic NLS

## TL;DR

This paper investigates the well-posedness, blowup, and scattering behavior of inhomogeneous cubic-quintic nonlinear Schrödinger equations in three dimensions, extending understanding of solutions with spatially varying coefficients under growth conditions.

## Contribution

It establishes local well-posedness, criteria for finite time blowup, and scattering results for ICQNLS with coefficients satisfying specific growth conditions, using Sobolev inequalities involving angular momentum operators.

## Key findings

- Proves local well-posedness for ICQNLS with certain coefficient conditions.
- Identifies conditions leading to finite time blowup.
- Demonstrates small data scattering and non-scattering scenarios.

## Abstract

In this paper we consider inhomogeneous cubic-quintic NLS in space dimension $d = 3$: $$ iu_t = -\Delta u + K_1(x)|u|^2u + K_2(x)|u|^4u. $$ 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 $x \neq 0$. To this end we use the Sobolev inequality for the functions $f \in H^n \,(n = 1, 2)$ such that $\||\mathbf L|^\ell f\|_{H^n} < \infty \,(\ell = 1, 2)$, where $\mathbf L$ is the angular momentum operator defined by $\mathbf L = x \times (-i\nabla)$.