Gradient Estimates for the doubly nonlinear diffusion equation on Complete Riemannian Manifolds

TL;DR

This paper studies the elliptic form of the doubly nonlinear diffusion equation $\Delta_{p}(u^{\gamma})+a u^{q}=0$ on complete Riemannian manifolds. It introduces a nonlinear transformation and applies Nash–Moser iteration to obtain Cheng–Yau type gradient estimates for positive solutions under a Ricci curvature lower bound, with Liouville-type results and Harnack inequalities as by-products. It addresses a gap in prior work by extending gradient estimates to the case $b>0$ and provides a partial extension to gradient bounds in that regime, along with a thorough treatment of the parameter constraints and the special case $a=0$. For $a=0$, the work yields additional gradient bounds for the transformed variable $v$ and includes a Caccioppoli-type inequality and Liouville-type conclusions, enriching the understanding of the elliptic doubly nonlinear diffusion on manifolds.

Abstract

We study the elliptic version of doubly nonlinear diffusion equations on a complete Riemannian manifold $(M,g)$. Through the combination of a special nonlinear transformation and the standard Nash-Moser iteration procedure, some Cheng-Yau type gradient estimates for positive solutions are derived. As by-products, we also obtain Liouville type results and Harnack's inequality. These results fill a gap in Yan and Wang (2018)\cite{YW}, due to the lack of one key inequality when $b=γ-\frac{1}{p-1}>0$, and provide a partial answer to the question that whether gradient estimates for the doubly nonlinear diffusion equation can be extended to the case $b>0$ .

Theorems & Definitions (37)

• Theorem A
• Theorem B
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Remark 1.1
• Corollary 1.5
• Corollary 1.6
• Definition 2.1