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

Let (MN,g,e−fdv) be a complete smooth metric measure space with ∞-Bakry–Émery Riccitensor bounded from below. We derive elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation (Δf−∂∂t)u(x, t)+q(x, t)uα(x, t)=0, where (x,t)∈MN×(−∞,∞) and α is an arbitrary constant. As Applications we prove a Liouville-type theorem for positive ancient solutions and Harnack-type inequalities for positive bounded solutions

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