Gradient estimates for u_t=ΔF(u) on manifolds and some Liouville-type theorems

2008/05/23 by Xiangjin Xu, Xu, Xiangjin
Computer Science · Mathematics · #35K05 #53C21 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.0805.3676

openalex publication_date 2008/05/23 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: u_t=ΔF(u), with F'(u) > 0, on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equation (PME): u_t=Δ(u^p), pgt;0, and obtain localized Hamilton-type gradient estimates for FDE and PME in a larger range of p than that for Aronson-Bénilan estimate, Harnack inequalities and Cauchy problems in the literature. Applying the localized gradient estimates for FDE and PME, we prove some Liouville-type theorems for positive global solutions of FDE and PME on noncompact complete manifolds with nonnegative Ricci curvature, generalizing Yaus celebrated Liouville theorem for positive harmonic functions.

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