On Existence and Uniqueness of Viscosity Solutions for Second Order Fully Nonlinear PDEs with Caputo time fractional derivatives

TL;DR

This paper establishes the existence and uniqueness of viscosity solutions for second order fully nonlinear PDEs with Caputo fractional time derivatives, expanding the theoretical framework for such fractional PDEs.

Contribution

It provides the first rigorous proof of existence and uniqueness for these fractional PDEs using comparison principles and Perron's method.

Findings

Proved comparison principle for fractional PDEs

Established existence and uniqueness of viscosity solutions

Extended viscosity solution theory to fractional derivatives

Abstract

Initial-boundary value problems for second order fully nonlinear PDEs with Caputo time fractional derivatives of order less than one are considered in the framework of viscosity solution theory. Associated boundary conditions are Dirichlet and Neumann, and they are considered in the strong sense and the viscosity sense, respectively. By a comparison principle and Perron's method, unique existence for the Cauchy-Dirichlet and Cauchy-Neumann problems are proved.