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 equationut=Δu−∇ϕ⋅∇u−aulogu−qu in M×[0,τ], where a∈R, ϕ is a C2-smooth function and q=q(x,t) is a function, which generalizes many previous well-known gradient estimate results.