Address: School of Mathematics and Physics, University of Science and Technology Beijing, Xueyuan Road 30, Haidian, Beijing 100083, China
E-mail: lma17@ustb.edu.cn
Abstract: In this paper, we consider two typical problems on a locally finite connected graph. The first one is to study the Bochner formula for the Laplacian operator on a locally finite connected graph. The other one is to obtain global nontrivial nonnegative solution to porous-media equation via the use of Aronson-Benilan argument. We use the curvature dimension condition to give a characterization two point graph. We also give a porous-media equation criterion about stochastic completeness of the graph. There is not much work in the direction of the study of nonlinear heat equations on locally finite connected graphs.
AMSclassification: primary 05C50; secondary 58J35, 53Cxx, 35Jxx, 68R10.
Keywords: Bochner formula, heat equation, global solution, stochastic completeness, porous-media equation, McKean type estimate.
DOI: 10.5817/AM2022-3-177