Li–Yau type estimates for a nonlinear parabolic equation on complete manifolds

Let ( M , g ) be a complete noncompact Riemannian manifold with the m-dimensional Bakry–Émery Ricci curvature bounded below. In this paper, we give a local Li–Yau type gradient estimate for the positive solutions to a general nonlinear parabolic equation u t = Δ u − ∇ ϕ ⋅ ∇ u − a u log u − q u in M × [ 0 , τ ] , where a ∈ R , ϕ is a C 2 -smooth function and q = q ( x , t ) is a function, which generalizes many previous well-known gradient estimate results.