Local Hamilton type Gradient Estimates and Harnack Inequalities for Nonlinear Parabolic Equations on Riemannian Manifolds

Abstract

In this paper, we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation

$${u_t}(x,t) = \Delta u(x,t) + au(x,t)\ln \,u(x,t) + b{u^\alpha }(x,t),$$

on M × (−∞, ∞) with α ∈ R, where a and b are constants. As application, the Harnack inequalities are derived.