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

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