Gradient Estimates and Harnack Inequality for a Nonlinear Parabolic Equation on Complete Manifolds

Abstract

Let $M$ be a noncompact complete Riemannian manifold. In this paper, we consider the following nonlinear parabolic equation on $M$

$\begin{aligned} u_t(x,t)=\Delta u(x,t) + a u(x,t)\ln u(x,t) + bu^{\alpha }(x,t). \end{aligned}$

We prove a Li–Yau type gradient estimate for positive solutions to the above equation; as an application, we also derive the corresponding Harnack inequality. These results generalize the corresponding ones proved by Li (J Funct Anal 100:233–256, 1991).

Keywords

Gradient estimate / Ricci curvature / Harnack inequality / Nonlinear parabolic equation