Finite time/Infinite time blow-up behaviors for the inhomogeneous nonlinear Schrödinger equation. (July 2023)

• Record Type: Journal Article
• Authors: Bai, Ruobing; Li, Bing
• Is Part Of: Nonlinear analysis. Volume 232(2023)
• Publication Date: 2023-07
• Subjects: primary 35Q55 -- secondary 35B44; Inhomogeneous nonlinear Schrödinger equation -- Blow-up -- Localized virial identity; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2023.113266
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this work, we consider the following focusing inhomogeneous nonlinear Schrödinger equation i ∂ t u + Δ u + | x | − b | u | p u = 0, ( t, x ) ∈ R × R N with 0 < b < min { 2, N } and 4 − 2 b N < p < 4 − 2 b N − 2 . Assume that u 0 ∈ H 1 ( R N ) and beyond the ground state threshold, then we prove the following two statements, (1) when 4 − 2 b N < p < min { 4 N, 4 − 2 b N − 2 }, or p = 4 N when b ∈ ( 0, 4 N ), then the corresponding solution blows up in finite time; (2) when 4 N < p < 4 − 2 b N − 2, we prove the finite or infinite time blow-up. Moreover, we can further obtain a precise lower bound of infinite time blow-up rate, that is sup t ∈ [ 0, T ] ‖ ∇ u ( t ) ‖ L 2 ≳ T κ, for some κ > 0 . To our knowledge, the statement (1) establishes the first finite time blow-up result for this equation in the intercritical case when the initial data u 0 does not have finite variance and is non-radial. The statement (2) gives the first result for the infinite time blow-up rate for this equation.