Li-Yau type and Souplet-Zhang type gradient estimates of a parabolic equation for the V-Laplacian

In this paper, we study a parabolic equation concerning V-Laplacian on complete Riemannian manifold M:(ΔV−q(x,t)−∂t)u(x,t)=A(u(x,t)) on M×[0,T], where V is a vector field on M and ΔV⋆:=Δ⋆+〈V,∇⋆〉. We give a gradient estimate of Li-Yau type for positive solutions of this equation. As corollary, we obtain a new gradient estimate in the case that A(u)=aulog⁡u. We also prove a Souplet-Zhang type gradient estimate for positive bounded solutions of this equation. As corollary, we obtain a gradient estimate for positive bounded solution of(ΔV−q(x,t)−∂t)u(x,t)=a(u(x,t))β,β>1.