Title: Nonlocal heat equations in the Heisenberg group,
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.