Blowup of H1 solutions for a class of the focusing inhomogeneous nonlinear Schrödinger equation. (September 2018)

• Record Type: Journal Article
• Authors: Dinh, Van Duong
• Is Part Of: Nonlinear analysis. Volume 174(2018)
• Page Start: 169
• Page End: 188
• Publication Date: 2018-09
• Subjects: 35B44 -- 35Q55; Inhomogeneous nonlinear Schrödinger equation -- Blowup -- Virial identity -- Gagliardo–Nirenberg inequality; 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.2018.04.024
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider a class of focusing inhomogeneous nonlinear Schrödinger equation i ∂ t u + Δ u + | x | − b | u | α u = 0, u ( 0 ) = u 0 ∈ H 1 ( R d ), with 0 < b < min { 2, d } and α ⋆ ≤ α < α ⋆ where α ⋆ = 4 − 2 b d and α ⋆ = 4 − 2 b d − 2 if d ≥ 3 and α ⋆ = ∞ if d = 1, 2 . In the mass-critical case α = α ⋆, we prove that if u 0 has negative energy and satisfies either x u 0 ∈ L 2 or u 0 is radial with d ≥ 2, then the corresponding solution blows up in finite time. Moreover, when d = 1, we prove that if the initial data (not necessarily radial) has negative energy, then the corresponding solution blows up in finite time. In the mass and energy intercritical case α ⋆ < α < α ⋆, we prove the finite time blowup for radial negative energy initial data as well as the finite time blowup below ground state for radial initial data in dimensions d ≥ 2 . This result extends the one of Farah in (J. Evol. Equ. 16: 193-208, 2016) where the blowup below ground state was proved for data in the virial space H 1 ∩ L 2 ( | x | 2 d x ) with d ≥ 1 .