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=Δ_$\mathrm{b}$u+au\log u+bu, \end{equation*} on $M\times [0,+\mathrm{\infty })$, 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 ${\mathrm{\Delta }}_{\mathrm{b}}u+au\log u=0$.