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Gradient Estimates for Some Semi-Linear Hypoelliptic Equations |
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| Authors: | Bin Qian Li Chen |
| Institution: | 1. School of Mathematics and Statistics, Changshu Institute of Technology, Changshu, 215500, China 2. School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai, 200433, China 3. Faculty of Mathematics and Computer Science, Hubei University, Hubei, 430062, China |
| Abstract: | Given some smooth vector fields X 1,X 2,…,X m on a compact manifold M, if they satisfy Hörmander’s condition, we establish global gradient estimates for the positive smooth solutions to the semi-linear hypoelliptic equations $$Lu+au\log u+bu=\partial_tu, \quad \mbox{on} \ M\times0,\infty) $$ and $$Lu+au\log u+bu=0, \quad \mbox{on} \ M, $$ where a,b are constants, and $L=\sum_{i}X_{i}^{2}-X_{0}$ . We partially generalize the results of Cao and Yau (Math. Z. 211:485–504, 1992). |
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