Gradient estimates for some f-heat equations driven by Lichnerowicz’s equation on complete smooth metric measure spaces

Given a complete, smooth metric measure space ( M , g , e - f d v ) with the Bakry–Émery Ricci curvature bounded from below, various gradient estimates for solutions of the following general f -heat equations u t = Δ f u + a u log u + b u + A u p + B u - q and u t = Δ f u + A e p u + B e - p u + D are studied. As by-product, we obtain some Liouville-type theorems and Harnack-type inequalities for positive solutions of several nonlinear equations including the Schrödinger equation, the Yamabe equation, and Lichnerowicz-type equations as special cases.