Gradient estimation of a generalized non-linear heat type equation along Super-Perelman Ricci flow on weighted Riemannian manifolds

TL;DR

This paper derives a Li-Yau-type gradient estimate for positive solutions of the generalized nonlinear parabolic equation $(\partial_t-\Delta_f)u = A(u)p(x,t) + B(u)q(x,t) + \mathcal{G}(u)$ on complete weighted Riemannian manifolds evolving under the $(k,m)$-super Perelman-Ricci flow $\partial_t g+2\,Ric_f^m \ge -2k g$. By introducing $h=\log(u/\delta)$ and $w=|\nabla h|^2/(1-h)^2$, the authors obtain a differential inequality for $w$ via the weighted Bochner formula and curvature-flow bounds, incorporating nonlinear terms $A,B,\mathcal{G}$ and their derivatives. This yields a local gradient bound for $|\nabla u|/u$ in terms of geometric data, the nonlinear data, and the cutoff radius, plus a parabolic Harnack-type inequality and a Liouville-type result as applications. The results generalize previous gradient estimates along Ricci-type flows to the $(k,m)$-super Perelman-Ricci flow, providing dimension-free and global implications for static and evolving weighted manifolds with nonlinear forcing terms.

Abstract

In this article we derive gradient estimation for positive solution of the equation \begin{equation*} (\partial_t-Δ_f)u = A(u)p(x,t) + B(u)q(x,t) + \mathcal{G}(u) \end{equation*} on a weighted Riemannian manifold evolving along the $(k,m)$ super Perelman-Ricci flow \begin{equation*} \frac{\partial g}{\partial t}(x,t)+2Ric_f^m(g)(x,t)\ge -2kg(x,t). \end{equation*} As an application of gradient estimation we derive a Harnack type inequality along with a Liouville type theorem.

Theorems & Definitions (18)

• Lemma 2.1
• Lemma 2.2
• Lemma 3.1
• Remark 3.1
• proof : Proof of Lemma \ref{['lemma3.1']}
• Lemma 3.2
• proof
• Lemma 3.3: Taheri
• Theorem 3.1
• proof