Neumann gradient estimate for nonlinear heat equation under integral Ricci curvature bounds

In this paper, we consider a Li-Yau gradient estimate on the positive solution to the following nonlinear parabolic equation

$$\frac{\mathrm{\partial }}{\mathrm{\partial }t}f=\mathrm{\Delta }f+af(\ln f{)}^p$$

with Neumann boundary conditions on a compact Riemannian manifold satisfying the integral Ricci curvature assumption, where $p\ge 0$ is a real constant. This contrasts Olivé's gradient estimate, which works mainly for the heat equation rather than nonlinear parabolic equations and the result can be regarded as a generalization of the Li-Yau [P. Li, S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153–201] and Olivé [X. R. Olivé, Neumann Li-Yau gradient estimate under integral Ricci curvature bounds, Proc. Amer. Math. Soc., 147 (2019), 411–426] gradient estimates.