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

TL;DR

This paper establishes Bernstein type gradient estimates for systems of weighted local heat equations with potentials on Riemannian manifolds, addressing various potential types and manifold conditions, and partially resolving a problem posed by Bhattacharyya et al.

Contribution

It provides new gradient estimates for heat systems with potentials on weighted manifolds, expanding previous results to multiple potential types and manifold settings.

Findings

Derived gradient estimates for linear and exponential potentials

Addressed static and evolving manifold cases

Partially resolved a problem from prior literature

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}.