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 = {https://global-sci.com/article/88150/gradient-estimates-for-a-nonlinear-heat-equation-under-finsler-geometric-flow} }