The Cauchy problem for nonlinear Schrödinger equations in modulation spaces. (August 2015)

• Record Type: Journal Article
• Authors: Ru, Shaolei; Chen, Jiecheng
• Is Part Of: Nonlinear analysis. Volume 24(2015)
• Page Start: 83
• Page End: 97
• Publication Date: 2015-08
• Subjects: Nonlinear Schrödinger equation -- Global well-posedness -- Modulation spaces; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2015.01.005
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: We consider the nonlinear Schrödinger equation ( ∗ ) { i u t + Δ u ± f ( u ) = 0, f ( u ) = | u | 2 m u n, m, n ∈ N, ( x, t ) ∈ R N × R u ( 0, x ) = u 0 ( x ) . We give a representation of the unique solution of the Cauchy problem( ∗ ) existed in C ( 0, T ∗ ; M p, 1 ) . Moreover, by this representation, we obtain that there exists a constant B independent of p, N such that for any initial data ‖ u 0 ‖ M p ′, 1 < B ( N ( 1 / 2 − 1 / p ) − 1 ), 2 N N − 2 < p ⩽ 2 m + n + 1, N > 2, the Schrödinger equation( ∗ ) has a unique global solution u ∈ C ( − ∞, ∞ ; M p, 1 ) .