Entire solutions of the nonlinear equation \(\begin{aligned} u_t-\triangle _g u=u^p, \ \ x\in M, t\in {\mathbb {R}} \end{aligned}\) were studied, where M is an N dimensional complete Riemannian manifold equipped with the metric g and p is assumed to be greater than one. The first part of this paper is devoted to investigation of Liouville property of \(\begin{aligned} u_t-\triangle _gu=f(u), \ \ x\in M, t\in {\mathbb {R}} \end{aligned}\) on compact manifolds for general function f may or may not change sign. Secondly, we will turn to non-compact manifolds and prove a Liouville theorem of (0.1) under the assumptions of boundedness of the Ricci curvature from below, diffeomorphism of M with \({\mathbb {R}}^N\) and sub-criticality of p defined below. Finally, we also present simplified proofs of Yau’s theorem for harmonic function and Gidas-Spruck’s theorem for elliptic semilinear equation. Our proofs are based on Li-Yau type estimation for nonlinear equations.
