Long time dynamics and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with spatial growing nonlinearity

Abstract

We investigate the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation $i{\partial }_{t}u+\Delta u=-|x{|}^b|u{|}^{p-1}u$ in the radial Sobolev space $H_{\text{rad}}^1({\mathbb{R}}^N)$, 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 $N\ge 3$. 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.

41 pages, an error in the proof of the energy scattering has been fixed

References in corpus (1)

• Finite time/Infinite time blow-up behaviors for the inhomogeneous nonlinear Schrödinger equation