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

摘要

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 Mathematics Subject Classification (2010) 58J35 35K10 35K55 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (8) References1.Calabi, E.: An extension of E. Hopf’s maximum principle with an application to Riemannian goemetry. Duke Math. J. 25, 45–56 (1958)CrossRefMATHMathSciNet2.Cheng, S.Y., Yau, S.T.: Differential equations on Riemannian manifolds and their geometric applications. Commun. Pure Appl. Math. 28, 333–354 (1975)CrossRefMATHMathSciNet3.Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 34, 525–598 (1981)CrossRefMATHMathSciNet4.Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)MATH5.Li, P., Yau, S.T.: On the parabolic kernel of the Schrodinger operator. Acta Math. 156, 153–201 (1986)CrossRefMathSciNet6.Li, J.: Gradient estimate and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds. J. Funct. Anal. 100, 233–256 (1991)CrossRefMATHMathSciNet7.Ma, L.: Gradient estimates for a simple elliptic equation on complete non-compact manifolds. J. Funct. Anal. 241, 374–382 (2006)CrossRefMATHMathSciNet8.Yang, Y.: Gradient estimate for a nonlinear parabolic equation on Riemannian manifold. Proc. Am. Math. Soc. 136, 4095–4102 (2008)CrossRefMATH About this Article Title Gradient Estimates and Harnack Inequality for a Nonlinear Parabolic Equation on Complete Manifolds Journal Communications in Mathematics and Statistics Volume 1, Issue 4 , pp 437-464 Cover Date2013-12 DOI 10.1007/s40304-014-0026-x Print ISSN 2194-6701 Online ISSN 2194-671X Publisher Springer Berlin Heidelberg Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Statistics, general Keywords Gradient estimate Ricci curvature Harnack inequality Nonlinear parabolic equation 58J35 35K10 35K55 Authors Jiaxian Wu (1) Yi-Hu Yang (2) Author Affiliations 1. Department of Mathematics, Tongji University, Shanghai , 200092, China 2. Department of Mathematics, Shanghai Jiao Tong University, Shanghai , 200240, China Continue reading... To view the rest of this content please follow the download PDF link above.