Long time dynamics and blow-up for the focusing inhomogeneous nonlinear Schr\'odinger equation with spatial growing nonlinearity
| Autor: | Dinh, Van Duong, Majdoub, Mohamed, Saanouni, Tarek |
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| Rok vydání: | 2021 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 41 pages, an error in the proof of the energy scattering has been fixed |
| Databáze: | arXiv |
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