Aronson–Bénilan estimates for the fast diffusion equation under the Ricci flow

Abstract We study the fast diffusion equation with a linear forcing term, (0.1) ∂ u ∂ t = div ( | u | p − 1 ∇ u ) + R u , under the Ricci flow on a complete manifold M such that M × R 2 has bounded curvature and nonnegative isotropic curvature, where 0 p 1 and R = R ( x , t ) is the evolving scalar curvature of M at time t . We prove Aronson–Benilan and Li–Yau–Hamilton type differential Harnack estimates for positive solutions of (0.1) . In addition, we use similar method to prove certain Li–Yau–Hamilton estimates for the heat equation and conjugate heat equation which extend those obtained in Cao and Hamilton (2009), Cao (2008), and Kuang and Zhang (2008) to the noncompact setting.