elliptic gradient estimates for a parabolic equation with v-laplacian and applications

In this paper, we establish a local elliptic gradient estimate for positive bounded solutions to a parabolic equation concerning the V-Laplacian (Formula presented) on an n-dimensional complete Riemannian manifold with the Bakry-Émery Ricci curvature Ric_V bounded below, which is weaker than the m-Bakry- Émery Ricci curvature (Formula presented) bounded below considered by Chen and Zhao (2018). As applications, we obtain the local elliptic gradient estimates for the cases that F(u) = au ln u and au^y. Moreover, we prove parabolic Liouville theorems for the solutions satisfying some growth restriction near infinity and study the problem about conformal deformation of the scalar curvature. In the end, we also derive a global Bernstein-type gradient estimate for the above equation with F(u) = 0.