gradient estimates and liouville theorems for nonlinear parabolic equations on noncompact riemannian manifolds

let m 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)u^α(x, t) = 0

on 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].