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-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) +2Ricfᵐ (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.