Gradient estimates for a weighted nonlinear parabolic equation

Let (Mn, g, e- f dv) be a complete smooth metric measure space. We prove elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation .f -.. t u(x, t) + q(x, t)u(x, t) + au(x, t)(ln u(x, t))a = 0, where (x, t). M x(-8,8) and a, a are arbitrary constants. Under the assumption that the 8-Bakry-Emery Ricci curvature is bounded from below, we obtain a local elliptic (Hamilton's type and Souplet-Zhang's type) gradient estimates to positive smooth solutions of this equation.