TY - JOUR
T1 - Gradient Estimates for a Nonlinear Heat Equation Under Finsler-Geometric Flow
AU - Zeng , Fanqi
JO - Journal of Partial Differential Equations
VL - 1
SP - 17
EP - 38
PY - 2020
DA - 2020/03
SN - 33
DO - http://doi.org/10.4208/jpde.v33.n1.2
UR - https://global-sci.org/intro/article_detail/jpde/15801.html
KW - Gradient estimate, nonlinear heat equation, Harnack inequality, Akbarzadeh's Ricci tensor, Finsler-geometric flow.
AB - <p style="text-align: justify;">This paper considers a compact Finsler manifold $(M^n, F(t), m)$ 
evolving under a Finsler-geometric flow and establishes global gradient 
estimates for positive solutions of the following nonlinear heat 
equation <br/>$$\partial_{t}u(x,t)=\Delta_{m} u(x,t),~~~~~~~~~~(x,t)\in M\times[0,T],$$</p><p style="text-align: justify;">where
 $\Delta_{m}$ is the Finsler-Laplacian. By integrating the gradient 
estimates, we derive the corresponding Harnack inequalities. Our results
 generalize and correct the work of S. Lakzian, who established similar 
results for the Finsler-Ricci flow. Our results are also natural 
extension of similar results on Riemannian-geometric flow, previously 
studied by J. Sun.&nbsp; Finally, we give an application to the 
Finsler-Yamabe flow.</p>