Fractional-Parabolic Systems

TL;DR

This paper develops a theoretical framework for solving linear evolution systems with fractional derivatives, extending classical parabolic systems and constructing the Green matrix using subordination identities.

Contribution

It introduces a new approach to analyze fractional parabolic systems by constructing the Green matrix via subordination, generalizing existing results for fractional diffusion equations.

Findings

Constructed the Green matrix for fractional parabolic systems.

Proved uniqueness of solutions using operator-differential equations.

Extended classical parabolic theory to fractional derivatives.

Abstract

We develop a theory of the Cauchy problem for linear evolution systems of partial differential equations with the Caputo-Dzrbashyan fractional derivative in the time variable $t$. The class of systems considered in the paper is a fractional extension of the class of systems of the first order in $t$ satisfying the uniform strong parabolicity condition. We construct and investigate the Green matrix of the Cauchy problem. While similar results for the fractional diffusion equations were based on the H-function representation of the Green matrix for equations with constant coefficients (not available in the general situation), here we use, as a basic tool, the subordination identity for a model homogeneous system. We also prove a uniqueness result based on the reduction to an operator-differential equation.