@article{Liu2024, 
author = {Hao-Yue Liu and Wei Zhang},
title = {Neumann gradient estimate for nonlinear heat equation under integral Ricci curvature bounds},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {2},
pages = {3881-3894},
keywords = {nonlinear parabolic equation, Li-Yau gradient estimate, integral Ricci curvature, Neumann boundary condition},
url = {https://www.sciopen.com/article/10.3934/math.2024191},
doi = {10.3934/math.2024191},
abstract = {In this paper, we consider a Li-Yau gradient estimate on the positive solution to the following nonlinear parabolic equation         ∂          ∂      t        f  =  Δ  f  +  a  f  (  ln  ⁡  f      )          p      with Neumann boundary conditions on a compact Riemannian manifold satisfying the integral Ricci curvature assumption, where    p  ≥  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.}
}