# Subordination principle for space-time fractional evolution equations and some applications

## TL;DR

This paper develops subordination formulae for space-time fractional evolution equations with Caputo derivatives, providing integral representations and applications to fractional diffusion problems, advancing analytical tools in fractional PDEs.

## Contribution

It introduces new subordination formulae for fractional evolution equations with Caputo derivatives and explores their properties and applications.

## Key findings

- Derived subordination formulae involving probability density functions.
- Established properties of the subordination kernels.
- Applied formulae to multi-dimensional fractional diffusion equations.

## Abstract

The abstract Cauchy problem for the fractional evolution equation with the Caputo derivative of order $\beta\in(0,1)$ and operator $-A^\alpha$, $\alpha\in(0,1)$, is considered, where $-A$ generates a strongly continuous one-parameter semigroup on a Banach space. Subordination formulae for the solution operator are derived, which are integral representations containing a subordination kernel (a scalar probability density function) and a $C_0$-semigroup of operators. Some properties of the subordination kernel are established and representations in terms of Mainardi function and L\'evy extremal stable densities are derived. Applications of the subordination formulae are given with a special focus on the multi-dimensional space-time fractional diffusion equation for some special values of the parameters.