A nonlinear Li-Yau inequality and its consequences

We prove a sharp nonlinear version of the celebrated Li-Yau inequality for positive solutions of the normalized parabolic $p$-Laplacian equation $u_t = Δu + (p-2)|\nabla u|^{-2}\nabla^2u(\nabla u,\nabla u)$, $1<p<\infty$, on a closed Riemannian manifold with nonnegative Ricci curvature, the equation