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

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}.