On the absence of global weak solutions for a nonlinear time-fractional Schr\"odinger equation

TL;DR

This paper investigates a nonlinear time-fractional Schrödinger equation with a singular potential, establishing conditions under which no global weak solutions exist, highlighting limitations in the equation's solvability.

Contribution

It provides new criteria for the nonexistence of global weak solutions for a fractional Schrödinger equation with a singular logarithmic potential.

Findings

No global weak solutions exist under certain conditions.

The test function method effectively demonstrates nonexistence.

Results highlight limitations in solving fractional Schrödinger equations with singular potentials.

Abstract

In this paper, an initial value problem for a nonlinear time-fractional Schr\"odinger equation with a singular logarithmic potential term is investigated. The considered problem involves the left/forward Hadamard-Caputo fractional derivative with respect to the time variable. Using the test function method with a judicious choice of the test function, we obtain sufficient criteria for the absence of global weak solutions.