Differential Harnack inequalities for semilinear parabolic equations on Riemannian manifolds II: Integral curvature condition

We present a unified method for deriving differential Harnack inequalities for positive solutions to semilinear parabolic equations on compact manifolds and complete Riemannian manifolds, subject to an integral curvature condition. Specifically, we obtain the differential Harnack inequalities by solving a related system of ordinary differential equations. In addition to the case of scalar equations, we also establish an elliptic estimate for the heat flow under the same condition, which is a novel result for both harmonic map and heat equations. Many of the results presented here are nearly sharp, meaning they are sharp under the assumption of Ricci nonnegativity.