The Cauchy problem for the Hartree type equation in modulation spaces. (January 2016)

• Record Type: Journal Article
• Authors: Bhimani, Divyang G.
• Is Part Of: Nonlinear analysis. Volume 130(2015)
• Page Start: 190
• Page End: 201
• Publication Date: 2016-01
• Subjects: primary 35Q55 42B35 -- secondary 35A01; Hartree equation -- Well-posedness -- Modulation spaces; 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.2015.10.002
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the Cauchy problem for Hartree equation with cubic convolution nonlinearity F ( u ) = ( K ∗ | u | 2 ) u under a specified condition on potential K with Cauchy data in modulation spaces M p, q ( R d ) . We establish global well-posedness results in M 1, 1 ( R d ) when K ( x ) = λ | x | − γ ( λ ∈ R, 0 < γ < min { 2, d / 2 } ) ; in M p, q ( R d ) ( 1 ≤ q ≤ min { p, p ′ } where p ′ is the Hölder conjugate of p ∈ [ 1, 2 ] ) when K is in Fourier algebra F L 1 ( R d ), and local well-posedness result in M p, 1 ( R d ) ( 1 ≤ p ≤ ∞ ) when K ∈ M 1, ∞ ( R d ) .