The nonexistence of global solutions for a time fractional nonlinear Schrödinger equation without gauge invariance. (February 2017) - Zhang, Quanguo; Sun, Hong-Rui; Li, Yaning
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 .