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

Let \((M^{n},g)\) $(M^n,g)$ be a complete Riemannian manifold. We prove a space-time gradient estimates for positive solutions of nonlinear parabolic equations \(\begin{aligned} \partial _{t}u(x,t)=\Delta u(x,t)-p(x,t)A(u(x,t))-q(x,t) ( u(x,t))^{a+1}, \end{aligned}\) $$\begin{array}{c}\hfill {\partial }_{t}u(x,t)=\mathrm{\Delta }u(x,t)-p(x,t)A(u(x,t))-q(x,t){(u(x,t))}^{a+1},\end{array}$$ on geodesic balls B(o, r) in M with \(0<r\le 1\) $0<r\le 1$ for \(s>\frac{n}{2}\) $s>\frac{n}{2}$ when integral Ricci curvature k(p, 1) is small enough. By integrating the gradient estimates in space-time we derive the corresponding Harnack inequalities.