# Nonlocal heat equations in the Heisenberg group

## TL;DR

This paper investigates nonlocal heat equations in the Heisenberg group, demonstrating that solutions exhibit asymptotic behavior similar to classical heat equations and establishing convergence of rescaled solutions to classical heat solutions.

## Contribution

It introduces a nonlocal diffusion model in the Heisenberg group and proves asymptotic equivalence and convergence results compared to classical heat equations.

## Key findings

- Solutions have the same asymptotic behavior as classical heat equations.
- Rescaled solutions of nonlocal problems converge uniformly to classical heat solutions.
- The spherical transform is used to analyze the problem.

## Abstract

We study the following nonlocal diffusion equation in the Heisenberg group $\mathbb{H}_n$, \[ u_t(z,s,t)=J\ast u(z,s,t)-u(z,s,t), \] where $\ast$ denote convolution product and $J$ satisfies appropriated hypothesis. For the Cauchy problem we obtain that the asymptotic behavior of the solutions is the same form that the one for the heat equation in the Heisenberg group. To obtain this result we use the spherical transform related to the pair $(U(n),\mathbb{H}_n)$. Finally we prove that solutions of properly rescaled nonlocal Dirichlet problem converge uniformly to the solution of the corresponding Dirichlet problem for the classical heat equation in the Heisenberg group.