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

Abstract

<abstract><p>In this paper, we consider a Li-Yau gradient estimate on the positive solution to the following nonlinear parabolic equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \frac{\partial}{\partial t}f = \Delta f+af(\ln f)^{p} $\end{document} </tex-math></disp-formula></p> <p>with Neumann boundary conditions on a compact Riemannian manifold satisfying the integral Ricci curvature assumption, where $ p\geq 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, <italic>Acta Math.,</italic> <bold>156</bold> (1986), 153–201] and Olivé [X. R. Olivé, Neumann Li-Yau gradient estimate under integral Ricci curvature bounds, <italic>Proc. Amer. Math. Soc.,</italic> <bold>147</bold> (2019), 411–426] gradient estimates.</p></abstract>