Liouville theorems and elliptic gradient estimates for a nonlinear parabolic equation involving the Witten Laplacian

In this paper, we establish local and global elliptic type gradient estimates for a nonlinear parabolic equation on a smooth metric measure space whose underlying metric and potential satisfy a (k,m)-super Perelman–Ricci flow inequality. We discuss a number of applications and implications including curvature free global estimates and some constancy and Liouville type results.