# An analytic semigroup generated by a fractional differential operator

## TL;DR

This paper investigates a space-fractional diffusion problem involving Caputo derivatives, proving the existence of regular solutions by demonstrating that a specific operator generates an analytic semigroup.

## Contribution

It establishes that the divergence of a positive function multiplied by the Caputo derivative generates an analytic semigroup, advancing the mathematical understanding of fractional diffusion operators.

## Key findings

- Proved the unique existence of regular solutions.
- Demonstrated the operator as a generator of an analytic semigroup.
- Extended semigroup theory to fractional differential operators.

## Abstract

We study a space-fractional diffusion problem, where the non-local diffusion flux involves the Caputo derivative of the diffusing quantity. We prove the unique existence of regular solutions to this problem by means of the semigroup theory. We show that the operator defined as divergence of product of a positive function with the Caputo derivative is a generator of analytic semigroup.