Gradient Estimates for a Nonlinear Parabolic Equation with Diffusion on Complete Noncompact Manifolds

The authors obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation:αu/αt=△u-b(x,t)u~σ on complete noncompact manifolds with Ricci curvature bounded from below,where 0<σ<1 is a real constant,and b(x,t) is a function which is C^2 in the x-variable and C^1 in the t-variable.