Regularity for a class of quasilinear degenerate parabolic equations in the Heisenberg group

We extend to the parabolic setting some of the ideas originated with Xiao Zhong's proof in [31] of the Holder regularity of $p-$harmonic functions in the Heisenberg group $\mathbb{H}^n$. Given a number $p\ge 2$, in this paper we establish the $C^{\infty}$ smoothness of weak solutions of a class of quasilinear PDE in $\mathbb{H}^n$ modeled on the equation $$?_t u = \sum_{i = 1}^{2n} X_i \bigg((1+|\nabla_0 u|^2)^{\frac{p-2}{2}} X_i u\bigg).$$