Harnack inequality for nonlinear parabolic equations under integral Ricci curvature bounds

Let $(M^n,g)$ be a complete Riemannian manifold. In this paper, we establish a space-time gradient estimates for positive solutions of nonlinear parabolic equations $${\mathrm{\partial }}_{t}u(x,t)=\mathrm{\Delta }u(x,t)+au(x,t)(\log u(x,t){)}^b+q(x,t)A(u(x,t)),$$ on geodesic balls $B(O,r)$ in $M$ with $0<r\le r$ for $p>\frac{n}{2}$ when integral Ricci curvature $k(p,1)$ is small enough. By integrating the gradient estimates, we find the corresponding Harnack inequalities.