Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Abstract

Let M be a complete noncompact Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation $$ \frac{\partial u}{\partial t} = \Delta _{f}u +au\,{\rm log}\, u + bu$$ on \({M \times [0, + \infty)}\), where a, b are two real constants, f is a smooth real-valued function on M and \({\Delta_f = \Delta - \nabla f \nabla}\). Under the assumption that the N-Bakry-Emery Ricci tensor is bounded from below by a negative constant, we obtain a gradient estimate for positive solutions of the above equation. As an application, we obtain a Harnack inequality and a Gaussian lower bound of the heat kernel of such an equation.