Local well-posedness of semilinear space-time fractional Schr\"odinger equation

TL;DR

This paper investigates the local well-posedness of the semilinear space-time fractional Schr"odinger equation by deriving explicit fundamental solutions and establishing decay and space-time estimates for mild solutions.

Contribution

It provides explicit fundamental solutions using Fox H-functions and establishes local well-posedness results for the equation.

Findings

Explicit fundamental solutions via Fox H-functions.

Decay estimates in L^s spaces.

Local well-posedness in specific Banach spaces.

Abstract

The semilinear space-time fractional Schr\"odinger equation is considered. First, we give the explicit form for the fundamental solutions by using the Fox $H$-functions in order to to establish some $L^s$ decay estimates. After that, we give some space-time estimates for the mild solutions from which the local well-posedness is derived on some proper Banach space.