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

Abstract

In this paper, we consider a Li-Yau gradient estimate on the positive solution to the following nonlinear parabolic equation $ \frac{\partial}{\partial t}f = \Delta f+af(\ln f)^{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, 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.

Keywords

• nonlinear parabolic equation
• li-yau gradient estimate
• integral ricci curvature
• neumann boundary condition