Bakry-Emery型Ricci曲率下两类抛物方程正解的梯度估计  

Gradient Estimates of the Positive Solutions to Two Parabolic Equations with the Bakry-Emery Ricci Curvature

作　　者：杨琼 Qiong YANG(School of Mathematics and Statistics,Wuhan University,Wuhan 430072,P.R.China)

机构地区：[1]武汉大学数学与统计学院,武汉430072

出　　处：《数学学报（中文版）》2022年第3期461-474,共14页Acta Mathematica Sinica：Chinese Series

摘　　要：本文考虑完备黎曼流形上,在Bakry-Emery型Ricci曲率有下界的条件下两类抛物方程■u/■t=△Vu+au log u和(△V-■/■t)u(x,t)+p(x,t)u^(β)(x,t)+q(x,t)u(x,t)=0正解的梯度估计,这里α,β∈R,△V(•):=△+〈V,▽(•)〉.由于引入了△V,相应地,在梯度估计证明的过程中用V-Laplace比较定理代替Laplace比较定理.作为梯度估计的应用,导出了Harnack不等式和Liouville定理.In this paper,we obtain gradient estimates for positive solutions of two nonlinear parabolic equations as follows ■u/■t=△Vu+au log u and (△V-■/■t)u(x,t)+p(x,t)u^(β)(x,t)+q(x,t)u(x,t)=0 where α,β∈R,△V(•):=△+〈V,▽(•)〉on the complete Riemannian manifold with Bakry-Emery Ricci curvature bounded below.Due to the introduction of Av,the Laplacian comparison theorem is replaced by the V-Laplacian comparison theorem in the process of proving the gradient estimates.Applications of these estimates yield Harnack inequalities and Liouville type theorem.

关 键 词：梯度估计 抛物方程 Bakry-Emery型Ricci曲率 HARNACK不等式 LIOUVILLE定理 

分 类 号：O186.1[理学—数学]