Gradient estimates for a nonlinear diffusion equation on complete manifolds

Details Gradient estimates for a nonlinear diffusion equation on complete manifolds Gradient estimates for a nonlinear diffusion equation on complete manifolds

2010-03-13

arxiv.org/abs/1003.2670v1

Journal of Partial Differential Equations, 23(2010), 68-79

Mathematics

Differential Geometry

11 pages

Scientific paper

10.4208/jpde.v23.n1.4

Let $(M,g)$ be a complete non-compact Riemannian manifold with the $m$-dimensional Bakry-\'{E}mery Ricci curvature bounded below by a non-positive constant. In this paper, we give a localized Hamilton-type gradient estimate for the positive smooth bounded solutions to the following nonlinear diffusion equation \[ u_t=\Delta u-\nabla\phi\cdot\nabla u-au\log u-bu, \] where $\phi$ is a $C^2$ function, and $a\neq0$ and $b$ are two real constants. This work generalizes the results of Souplet and Zhang (Bull. London Math. Soc., 38 (2006), pp. 1045-1053) and Wu (Preprint, 2008).

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