Gradient estimates for a nonlinear diffusion equation on complete manifolds

Abstract

This paper deals with the gradient estimates of the Hamilton type for the positive solutions to the following nonlinear diffusion equation: ${u_t} = \Delta u + \nabla \phi \cdot \nabla u + a\left( x \right)u\ln u + b\left( x \right)u$ on a complete noncompact Riemannian manifold with a Bakry-Emery Ricci curvature bounded below by −K (K ≥ 0), where φ is a C ² function, a(x) and b(x) are C ¹ functions with certain conditions.

Keywords

Gradient estimate / Bakry-Emery Ricci curvature / Nonlinear diffusion equation

References

[1] Bakry D., Emery M.. Diffusion Hypercontractivites, Séminaire de Probabilités XIX, 1983/1984, Lect. Notes in Math, 1985, 1123: 177-206

[2] Hamilton R. S.. A matrix Harnack estimate for the heat equation. Comm. Anal. Geom., 1993, 1: 113-126

[3] Li P., Yau S. T.. On the parabolic kernel of the Schrödinger operator. Acta Math, 1986, 156: 153-201

[4] Li J. Y.. Gradient estimate and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds. J. Funct. Anal, 1991, 100: 233-256

[5] Li X. D.. Liouville theorem for symmeric diffusion operators on complete Riemannian manifolds. J. Math. Pure Appl, 2005, 84: 1295-1361

[6] Ma L.. Gradient estimates for a simple elliptic equation on non-compact complete manifolds. J. Funct. Anal, 2006, 241: 374-382

[7] Qian B.. Hamilton-type gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Acta Math. Sin, 2011, 27: 1071-1078

[8] Qian Z. M.. A comparison theorem for an elliptic operator. Potential Analysis, 1998, 8: 137-142

[9] Ruan Q. H.. Elliptic-type gradient estimate Schr¨odinger equations on noncompact manifolds. Bull. London Math. Soc, 2007, 39: 982-988

[10] Souplet P., Zhang Q. S.. Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Bull. London Math. Soc, 2006, 38: 1045-1053

[11] Wei G. F., Wylie W.. Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom, 2009, 83(2): 337-405

[12] Yang Y. Y.. Gradient estimate for a nonlinear parabolic equation on Riemannian manifold. Proc. Amer. Math. Soc, 2008, 136: 4095-4102