Gradient Estimates for a Weighted Γ-nonlinear Parabolic Equation Coupled with a Super Perelman-Ricci Flow and Implications

Abstract

This article studies a nonlinear parabolic equation on a complete weighted manifold where the metric and potential evolve under a super Perelman-Ricci flow. It derives elliptic gradient estimates of local and global types for the positive solutions and exploits some of their implications notably to a general Liouville type theorem, parabolic Harnack inequalities and classes of Hamilton type dimension-free gradient estimates. Some examples and special cases are discussed for illustration.

Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications

“…[16,19,51,48]). Further applications and results in this direction will be discussed in a forthcoming paper (see also [40,41]).…”

“…have been studied in detail in [9,17,40,41,46,47]. Yamabe type equations ∆u + p(x)u s + q(x)u = 0 are also of form (1.1) with a powerlike nonlinearity.…”

“…(1.13) For gradient estimates, Harnack inequalities and Liouville-type results in this and related contexts see Dung, Khanh and Ngo [16], Song and Zhao [36], Taheri [40,41], Wu [48] and the references therein.…”

Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials

“…Gradient estimates occupy a central place in geometric analysis with a huge scope of applications (see [2,8,15,19,25,32,33,34], [1,6,14,18,43,47,52,53,55,56,58,61,62] as well as [5,21,23,31,51,64] and the references therein). The estimates of interest in this paper fall under the category of Souplet-Zhang, Hamilton and Li-Yau types that were first formulated and proved for the linear heat equation on static manifolds in [25,47] and later extended by many authors to contexts including evolving manifolds and nonlinear equations.…”

“…where p, q are real exponents, a, b are sufficiently smooth functions and Γ ∈ C 2 (R, R) (see [9,20,52,53,60,61] and the references therein for gradient estimates and related results in this direction). Another class of equations that have been extensively studied and whose nonlinearity is in the form of superposition of power-like nonlinearities are the Yamabe equations (see, e.g., [6,13,24,30]).…”