Gradient Estimates for a Nonlinear Heat Equation Under Finsler-Geometric Flow (Journal of Partial Differential Equations, 2020)
"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.\u00a0 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.