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

Let \((M,J,\theta )\) $(M,J,\theta )$ be a complete noncompact pseudo-Hermitian manifold which satisfies the CR sub-Laplacian comparison property. We first obtain local subgradient estimates for positive solutions to the following nonlinear subparabolic equation: \(\begin{aligned} u_t=\Delta _bu+au\ln u+bu, \end{aligned}\) $$\begin{array}{c}\hfill u_t={\mathrm{\Delta }}_{b}u+aulnu+bu,\end{array}$$ on \(M\times [0,+\infty )\) $M\times [0,+\infty )$ , where a, b are two real constants. As a application, we derive a priori estimate for positive solutions to the subelliptic equation \(\Delta _bu+au\ln u=0\) ${\mathrm{\Delta }}_{b}u+aulnu=0$ . Secondly, we deform the pseudo-Hermitian form by the general CR flow on a complete noncompact pseudo-Hermitian 3-manifold. We show that if a pseudo-Hermitian form evolves, then we have a new local subgradient estimate for positive solutions to the equation \(u_t=\Delta _bu+au\ln u+bu\) $u_t={\mathrm{\Delta }}_{b}u+aulnu+bu$ .