Large time behaviour of the fractional heat equation associated with the Dunkl Laplacian

TL;DR

This paper studies the long-term behavior of solutions to the fractional heat equation linked with the Dunkl Laplacian, showing convergence to the fundamental solution for large times under certain initial conditions.

Contribution

It establishes the asymptotic convergence of solutions to the fractional Dunkl heat equation and extends the result to a nonlinear variant.

Findings

Solutions converge to the fundamental solution as time increases.

Convergence holds for initial data integrable with respect to the Dunkl measure.

Results apply to both linear and nonlinear fractional Dunkl heat equations.

Abstract

We consider the fractional heat equation associated with the Dunkl Laplacian and prove that the weak solutions to this equation converge to the fundamental solution as time becomes large, provided the initial data is an integrable function with respect to the associated measure. As an application, we also prove a similar result for the corresponding nonlinear equation.