Global existence and blowup for a class of the focusing nonlinear\n Schr "odinger equation with inverse-square potential

2017/11/13 by Van Duong Dinh, Dinh, Van Duong · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems

paper · pdf · doi:10.48550/arxiv.1711.04792

openalex publication_date 2017/11/13 · openalex created_date 2022/08/13 · openalex updated_date 2026/07/28

Abstract

We consider a class of the focusing nonlinear Schr "odinger equation with\ninverse-square potential \i
partial_t u +
Delta u -c|x|^-2u = - |u|^
alphaν,
quad u(0)=u₀
in H¹,
quad (t,x)
in
mathbbR
times
mathbbR^d, nwhere d\≥ 3, \(4)/(d)\≤ \α \≤ \(4)/(d-2) and c\≠ 0\nsatisfies c>-\λ(d):=-\(\(d-2)/(2)\)². In the mass-critical\ncase \α=\(4)/(d), we prove the global existence and blowup below\nground states for the equation with d\≥ 3 and c>-\λ(d). In the mass\nand energy intercritical case \(4)/(d)<\α<\(4)/(d-2), we prove the\nglobal existence and blowup below the ground state threshold for the equation.\nThis extends similar results of citeKillipMurphyVisanZheng and\n citeLuMiaoMurphy to any dimensions d\≥ 3 and a full range\nc>-\λ(d). We finally prove the blowup below ground states for the\nequation in the energy-critical case \α=\(4)/(d-2) with d\≥ 3 and\nc>-\(d²+4d)/((d+2)²) \λ(d).\n

Cited by

• Damping Effects on Global Existence and Scattering for an Inhomogeneous NLS Equation with Inverse-Square Potential

Related

• Global existence and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential
• On the blowup for the L²-critical focusing nonlinear Schrödinger equation in higher dimensions below the energy class
• Blow-up for the nonlinear Schrödinger equation with combined nonlinearities
• On blowup solutions to the focusing intercritical nonlinear fourth-order Schrödinger equation
• Blow-up for the 3D intercritical inhomogeneous NLS with inverse-square potential