Gradient Estimates of Li Yau Type for a General Heat Equation on Riemannian Manifolds

In this paper, we consider gradient estimates on complete noncompact Riemannian manifolds (M, g) for the following general heat equation ut = ∆V u+ au log u+ bu where a is a constant and b is a differentiable function defined onM×[0,∞). We suppose that the Bakry-Émery curvature and the N -dimensional Bakry-Émery curvature are bounded from below, respectively. Then we obtain the gradient estimate of Li-Yau type for the above general heat equation. Our results generalize the work of Huang-Ma ([4]) and Y. Li ([6]), recently. 1.. Introduction Recently, the weighted Laplacian on smooth metric measure spaces has been attracted by many researchers. Recall that a triple (M, g, e−fdv) is called a smooth metric measure space if (M, g) is a Riemannian manifold, f is a smooth function on M and dv is the volume form with respect to g. On smooth metric measure spaces, the weighted Laplace operator is defined by ∆f · := ∆ · − 〈∇f,∇·〉 where ∆ is the Laplace operator on M . On (M, g, e−fdv), the Bakry-Émery curvature Ricf and the N -dimensional Bakry-Émery curvarute Ricf are defined by Ricf := Ric + Hess f, Ricf := Ricf − 1 N ∇f ⊗∇f where Ric, Hess f are the Ricci curvature and the Hessian of f on M , respectively. An important generalization of the weighted Laplace operator on Riemannian manifolds is the following operator ∆V · := ∆ ·+ 〈V,∇·〉 where∇ and ∆ are respectively the Levi-Civita connection and the Laplace-Beltrami operator with respect to g, V is a smooth vector field on M . In [1] and [6], the 2010 Mathematics Subject Classification: primary 58J35; secondary 35B53.