Subgradient estimates for a nonlinear subparablic equation on complete pseudo-Hermitian manifolds

Let $(M,J,\theta)$ be a complete pseudo-Hermitian manifold which satisfies the CR sub-Laplacian comparison property. In this paper, we derive the local subgradient estimates for positive solutions to the following nonlinear subparabolic equation: \begin{equation*} u_t=\Delta_\mathrm{b}u+au\log u+bu, \end{equation*} on $M\times [0,+\infty)$, where $a,b\in \mathbb{R}$. As a application, we derive a priori estimate and a Harnack inequality for positive solutions to the subelliptic equation $\Delta_\mathrm{b}u+au\log u=0$.