The nonexistence of global solutions for a time fractional nonlinear Schrödinger equation without gauge invariance. (February 2017)

• Record Type: Journal Article
• Authors: Zhang, Quanguo; Sun, Hong-Rui; Li, Yaning
• Is Part Of: Applied mathematics letters. Volume 64(2017:Feb.)
• Page Start: 119
• Page End: 124
• Publication Date: 2017-02
• Subjects: Time fractional Schrödinger equation -- Nonexistence -- Non-gauge invariance; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2016.08.017
• Languages: English
• ISSNs: 0893-9659

Abstract

Abstract: In this paper, we study the following time fractional Schrödinger equation i α 0 C D t α u + △ u = λ | u | p, x ∈ R N, t ∈ [ 0, T ), where 0 < α < 1, i α denotes the principal value of i α, T > 0, λ ∈ C ∖ { 0 }, p > 1, u ( t, x ) is a complex-valued function, and 0 C D t α u denotes Caputo fractional derivative of order α . We prove that the problem admits no global weak solution with suitable initial data when 1 < p < 1 + 2 ∕ N by using the test function method, and also give some conditions which imply the problem has no global weak solution for every p > 1 .