Blow‐up versus global well‐posedness for the focusing INLS with inverse‐square potential. (19th September 2022)

• Record Type: Journal Article
• Authors: Deng, Mingming; Lu, Jing; Meng, Fanfei
• Other Names: Gao Feng guestEditor.; Feng Wenying guestEditor.; Zhang Xinguang guestEditor.; Ge Fudong guestEditor.
• Is Part Of: Mathematical methods in the applied sciences. Volume 46:Number 3(2023)
• Page Start: 3285
• Page End: 3293
• Publication Date: 2022-09-19
• Subjects: blow‐up -- global well‐posedness -- inverse‐square potential -- Virial identity; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.8690
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : We study the focusing inhomogeneous nonlinear Schrödinger equation with inverse‐square potential i ∂ t u + Δ u − a | x | 2 u + | x | − b | u | 2 u = 0, $$ i{\partial}_tu&amp;amp;#x0002B;\Delta u-\frac{a}{{\left&amp;amp;#x0007C;x\right&amp;amp;#x0007C;}&amp;amp;#x0005E;2}u&amp;amp;#x0002B;{\left&amp;amp;#x0007C;x\right&amp;amp;#x0007C;}&amp;amp;#x0005E;{-b}{\left&amp;amp;#x0007C;u\right&amp;amp;#x0007C;}&amp;amp;#x0005E;2u&amp;amp;#x0003D;0, $$ where a > − 1 4 $$ a&amp;gt;-\frac{1}{4} $$ and 0 < b < 1 $$ 0&amp;lt;b&amp;lt;1 $$ in dimension three. We extend the results of Campos and Cardoso to inhomogeneous nonlinear Schrödinger equation (INLS) with inverse‐square potential, and the proof is based on the method from Duyckaerts and Roudenko. Furthermore, our result compensates for the one of Campos and Guzmán, obtaining blow‐up versus global existence dichotomy for solutions beyond the threshold.