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.
作者
杨琼
Qiong YANG(School of Mathematics and Statistics,Wuhan University,Wuhan 430072,P.R.China)
出处
《数学学报（中文版）》
CSCD
北大核心
2022年第3期461-474,共14页
Acta Mathematica Sinica：Chinese Series
