Differential gradient estimates for nonlinear parabolic equations under integral Ricci curvature bounds

Let \((M^{n},g)\) be a complete Riemannian manifold. We prove a space-time gradient estimates for positive solutions of nonlinear parabolic equations

$$\begin{aligned} \partial _{t}u(x,t)=\Delta u(x,t)-p(x,t)A(u(x,t))-q(x,t) ( u(x,t))^{a+1}, \end{aligned}$$

on geodesic balls B(o, r) in M with \(0<r\le 1\) for \(s>\frac{n}{2}\) when integral Ricci curvature k(p, 1) is small enough. By integrating the gradient estimates in space-time we derive the corresponding Harnack inequalities.