A Souplet–Zhang type gradient estimate for the fast diffusion equation associated with the Witten Laplacian

Abstract

We establish a Souplet–Zhang type local gradient estimate for positive solutions $u=u(x,t)$ to the fast diffusion equation associated with the Witten Laplacian$\frac{\partial u}{\partial t}={\Delta }_V{u}^m,1-\frac{2}{N}<m<1$ on an n-dimensional Riemannian manifold $(X,g)$ when the N-Bakry–Émery Ricci curvature with $N\in [n,+\infty )$ is bounded from below by a non-positive constant. When the N-Bakry–Émery Ricci curvature is reduced to the Ricci curvature, our result refines the Souplet–Zhang type local gradient estimate by X. Zhu (2011) [10]. As an application, we prove a Liouville type theorem for positive ancient solutions to the fast diffusion equation associated with the Witten Laplacian on an n-dimensional non-compact Riemannian manifold $(X,g)$ with non-negative N-Bakry–Émery Ricci curvature with $N\in [n,+\infty )$.