Harnack inequality for nonlinear parabolic equations under integral Ricci curvature bounds

TL;DR

This paper derives Harnack inequalities for positive solutions of certain nonlinear parabolic equations on Riemannian manifolds with small integral Ricci curvature, extending classical results under curvature bounds.

Contribution

It establishes space-time gradient estimates and Harnack inequalities for nonlinear parabolic equations under integral Ricci curvature bounds, a novel extension of existing geometric analysis results.

Findings

Gradient estimates for solutions under integral Ricci curvature bounds

Harnack inequalities derived from gradient estimates

Results applicable to manifolds with small integral Ricci curvature

Abstract

Let $(M^{n},g)$ be a complete Riemannian manifold. In this paper, we establish a space-time gradient estimates for positive solutions of nonlinear parabolic equations $$\partial_{t}u(x,t)=\Delta u(x,t)+a u(x,t)(\log u(x,t))^b + q(x,t)A(u(x,t)),$$ on geodesic balls $B(O,r)$ in $M$ with $0<r\leq r$ for $p>\frac{n}{2}$ when integral Ricci curvature $k(p,1)$ is small enough. By integrating the gradient estimates, we find the corresponding Harnack inequalities.