Elliptic gradient estimates and Liouville theorems for a weighted nonlinear parabolic equation

Let(MN,g,e−fdv)beacompletesmoothmetricmeasurespacewith∞-Bakry–ÉmeryRiccitensorboundedfrombelow.Wederiveellipticgradientestimatesforpositivesolutionsofaweightednonlinearparabolicequation(Δf−∂∂t)u(x, t)+q(x, t)uα(x, t)=0,where(x,t)∈MN×(−∞,∞)andαisanarbitraryconstant.AsApplicationsweproveaLiouville-typetheoremforpositiveancientsolutionsandHarnack-typeinequalitiesforpositiveboundedsolution