Life span of blowing-up solutions to the Cauchy problem for a time-fractional Schrödinger equation

In this paper, we will consider the Cauchy problem for a time-fractional Schrödinger equation with Riemann–Liouville nonlinear fractional integral term. This class of equations have interesting applications for large systems of self-interactions, which allow us to use the fractional calculus techniques to investigate long range interactions and quantum processes. By utilizing the test function method and some important properties of fractional calculus, we give a blow-up result involving the criterion of verifying whether there is a global nontrivial weak solution. Then, by establishing some integral inequalities, we provide an upper bound estimate for the life span of the blowing-up solutions. Finally, a numerical example is presented to demonstrate the validity of our theoretical results. The obtained results generalize the previous ones, because the analogous problem with a time fractional derivative has not been studied so far.