Gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds

Let MM be a complete noncompact Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions of the nonlinear parabolic equation (Δ−∂∂t)u(x,t)+λ(x,t)uα(x,t)=0 on M×(−∞,0]M×(−∞,0]. We also obtain a theorem of Liouville type for positive solutions of this nonlinear equation. This paper extends the result of Souplet and Zhang [P. Souplet, Qi S. Zhang, Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds, Bull. Lond. Math. Soc. 38 (2006) 1045–1053].