TY - JOUR
T1 - Gradient Estimates for a Nonlinear Diffusion Equation on Complete Manifolds
JO - Journal of Partial Differential Equations
VL - 1
SP - 68
EP - 79
PY - 2010
DA - 2010/02
SN - 23
DO - http://doi.org/10.4208/jpde.v23.n1.4
UR - https://global-sci.org/intro/article_detail/jpde/5222.html
KW - Local gradient estimate
KW - nonlinear diffusion equation
KW - Bakry-Émery Ricci curvature
AB - <p>Let (M,g) be a complete non-compact Riemannian manifold with the m- dimensional Bakry-Émery 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=Δu-∇φ·∇u-au\log u-bu, where φ is a C^2 function, and a ≠ 0 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).</p>