Bernstein type gradient estimate for system of weighted local heat equations with potential term

TL;DR

This work develops Bernstein-type gradient estimates for a pair of local weighted heat equations with linear and exponential potentials on both static and evolving weighted Riemannian manifolds. Using the maximum principle and curvature-dominated Bochner-type calculations, it derives explicit gradient bounds of the form $|\nabla u|^2 \le \frac{C}{\chi^2 t}$ and $|\nabla v|^2 \le \frac{C}{\chi^2 t}$, with constants determined by initial data, potential parameters, and (where applicable) curvature bounds. The static-manifold results cover four cases (linear/exponential potentials) and rely on quantities $\Phi_0(K)$ or $\Psi_0(K,\xi)$, while the evolving-manifold results extend to local Ricci flow, introducing curvature-free terms and new constants $\Lambda_0$ and $\Gamma_0$ that govern the bounds. The findings partially address a program proposed by Bhattacharyya et al. and open avenues for future work on nonlinear systems such as Keller-Segel and extensions to non-Riemannian geometries.

Abstract

In this article we provide Bernstein type gradient estimates for two system of local weighted heat type equations with potentials on a weighted Riemannian manifold. We derive all possible cases considering linear potential, exponential potential, combining with static manifold and evolving manifold. This work partially resolved the problem raised by Bhattacharyya et al. in \cite{SB-1}.

Theorems & Definitions (11)

• Remark 1.1
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Lemma 3.1: Extension of Lemma 2.2 of SB-1
• proof
• Theorem 3.1
• proof
• Theorem 3.2