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

Chen Guo; Zhengce Zhang

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arXiv · 2504.08276

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

Chen Guo·Zhengce Zhang

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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 $b=\gamma-\frac{1}{p-1}>0$, and provide a partial answer to the question that whether gradient estimates for the doubly nonlinear diffusion equation can be extended to the case $b>0$ .

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Chen Guo, Zhengce Zhang. 2025-04-11. Gradient Estimates for the doubly nonlinear diffusion equation on Complete Riemannian Manifolds. https://arxiv.org/abs/2504.08276

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