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Volume 33, Issue 1
Gradient Estimates for a Nonlinear Heat Equation Under Finsler-Geometric Flow

Fanqi Zeng

DOI: 10.4208/jpde.v33.n1.2

J. Part. Diff. Eq., 33 (2020), pp. 17-38.

Published online: 2020-03

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  • Abstract

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
$$\partial_{t}u(x,t)=\Delta_{m} u(x,t),~~~~~~~~~~(x,t)\in M\times[0,T],$$

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.  Finally, we give an application to the Finsler-Yamabe flow.

  • Keywords

Gradient estimate, nonlinear heat equation, Harnack inequality, Akbarzadeh's Ricci tensor, Finsler-geometric flow.

  • AMS Subject Headings

53C44, 58J35, 53B40, 35K55

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

fanzeng10@126.com (Fanqi Zeng)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-33-17, author = {Zeng , Fanqi}, title = {Gradient Estimates for a Nonlinear Heat Equation Under Finsler-Geometric Flow}, journal = {Journal of Partial Differential Equations}, year = {2020}, volume = {33}, number = {1}, pages = {17--38}, abstract = {

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
$$\partial_{t}u(x,t)=\Delta_{m} u(x,t),~~~~~~~~~~(x,t)\in M\times[0,T],$$

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.  Finally, we give an application to the Finsler-Yamabe flow.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v33.n1.2}, url = {http://global-sci.org/intro/article_detail/jpde/15801.html} }
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 -

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
$$\partial_{t}u(x,t)=\Delta_{m} u(x,t),~~~~~~~~~~(x,t)\in M\times[0,T],$$

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.  Finally, we give an application to the Finsler-Yamabe flow.

Zeng , Fanqi. (2020). Gradient Estimates for a Nonlinear Heat Equation Under Finsler-Geometric Flow. Journal of Partial Differential Equations. 33 (1). 17-38. doi:10.4208/jpde.v33.n1.2
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