Global well-posedness of space-time fractional diffusion equation with Rockland operator on graded Lie group

TL;DR

This paper proves the global well-posedness and regularity of space-time fractional diffusion equations involving Rockland operators on graded Lie groups, covering various important examples like the Heisenberg group.

Contribution

It establishes the existence, uniqueness, and regularity of solutions for a broad class of fractional diffusion equations on graded Lie groups, extending previous results to more general operators.

Findings

Proved global well-posedness for the fractional diffusion equation.

Established regularity estimates for solutions.

Unified treatment of various examples including sub-Laplacian cases.

Abstract

In this article, we examine the general space-time fractional diffusion equation for left-invariant hypoelliptic homogeneous operators on graded Lie groups. Our study covers important examples such as the time-fractional diffusion equation, the space-time fractional diffusion equation when diffusion is under the influence of sub-Laplacian on the Heisenberg group, or general stratified Lie groups. We establish the global well-posedness of the Cauchy problem for the general space-time fractional diffusion equation for the Rockland operator on a graded Lie group in the associated Sobolev spaces. More precisely, we establish the existence and uniqueness results for both homogeneous and inhomogeneous fractional diffusion equations. In addition, we also develop some regularity estimates.