Li-Yau gradient bound for collapsing manifolds under integral curvature condition

Let $(\M^n, g_{ij})$ be a complete Riemammnian manifold. For some constants $p,\ r>0$, define $\displaystyle k(p,r)=\sup_{x\in M}r^2\left(\oint_{B(x,r)}|Ric^-|^p dV\right)^{1/p}$, where $Ric^-$ denotes the negative part of the Ricci curvature tensor. We prove that for any $p>\frac{n}{2}$, when $k(p,1)$ is small enough, certain Li-Yau type gradient bound holds for the positive solutions of the heat equation on geodesic balls $B(O,r)$ in $\M$ with $0<r\leq 1$. Here the assumption that $k(p,1)$ being small allows the situation where the manifolds is collapsing. Recall that in \cite{ZZ}, certain Li-Yau gradient bounds was also obtained by the authors, assuming that $|Ric^-|\in L^p(\M)$ and the manifold is noncollaped. Therefore, to some extent, the results in this paper and in \cite{ZZ} complete the picture of Li-Yau gradient bound for the heat equation on manifolds with $|Ric^-|$ being $L^p$ integrable, modulo sharpness of constants.