Long time dynamics and blow-up for the focusing inhomogeneous nonlinear Schr\"odinger equation with spatial growing nonlinearity

TL;DR

This paper studies the long-term behavior and blow-up phenomena of solutions to a focusing inhomogeneous nonlinear Schrödinger equation with spatially growing nonlinearity, establishing conditions for global existence, scattering, and blow-up.

Contribution

It introduces refined inequalities and analytical techniques to handle spatial growth, extending understanding of solution dynamics in inhomogeneous NLS equations.

Findings

Global existence and scattering in the inter-critical regime.

Existence of blow-up solutions in mass-critical and supercritical cases.

Development of refined Gagliardo-Nirenberg inequalities for inhomogeneous settings.

Abstract

We investigate the Cauchy problem for the focusing inhomogeneous nonlinear Schr\"odinger equation $i \partial_t u + \Delta u = - |x|^b |u|^{p-1} u$ in the radial Sobolev space $H^1_{\text{rad}}(\mathbb{R}^N)$, where $b>0$ and $p>1$. We show the global existence and energy scattering in the inter-critical regime, i.e., $p>\frac{N+4+2b}{N}$ and $p<\frac{N+2+2b}{N-2}$ if $N\geq 3$. We also obtain blowing-up solutions for the mass-critical and mass-supercritical nonlinearities. The main difficulty, coming from the spatial growing nonlinearity, is overcome by refined Gagliardo-Nirenberg type inequalities. Our proofs are based on improved Gagliardo-Nirenberg inequalities, the Morawetz-Sobolev approach of Dodson and Murphy, radial Sobolev embeddings, and localized virial estimates.