# On viscosity solutions of space-fractional diffusion equations of Caputo type

## TL;DR

This paper introduces a notion of viscosity solutions for space-fractional diffusion equations of Caputo type, proves their existence and uniqueness, and explores their stability and regularity properties.

## Contribution

It defines viscosity solutions for Caputo-type fractional diffusion equations and establishes their existence, uniqueness, and stability, advancing the theoretical understanding of such equations.

## Key findings

- Existence of viscosity solutions proved using Perron's method
- Uniqueness established via a maximum principle
- Solutions exhibit stability and basic regularity

## Abstract

We study a fractional diffusion problem in the divergence form in one space dimension. We define a notion of the viscosity solution. We prove existence of viscosity solutions to the fractional diffusion problem with the Dirichlet boundary values by Perron's method. Their uniqueness follows from a proper maximum principle. We also show a stability result and basic regularity of solutions.