Life span of solutions to a nonlocal in time nonlinear fractional Schrödinger equation

In this paper, we study the initial value problem for the nonlocal in time nonlinear Schrödinger equation iut+Δu=λJ0|tα|u|p,x∈RN,t>0,u(x,0)=f(x),x∈RN.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} iu_{t}+\Delta u = \lambda J^{\alpha}_{0\vert t} \vert u\vert^p, \quad x \in \mathbb{R}^N, \quad t > 0,\\ u(x,0) = f(x), \quad x \in \mathbb{R}^N.\quad \quad \end{aligned}$$\end{document}Using the test function method, we derive a blow-up exponent. Then based on integral inequalities, we estimate the life span of blowing-up solutions.