The nonexistence of global solutions for a time fractional nonlinear Schrödinger equation without gauge invariance

Abstract

In this paper, we study the following time fractional Schrödinger equation iα0CDtαu+△u=λ|u|p,x∈RN,t∈[0,T),where 0<α<1, iα denotes the principal value of iα, T>0, λ∈C∖{0}, p>1, u(t,x) is a complex-valued function, and 0CDtαu 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.