Gradient Estimates for a Nonlinear Heat Equation Under Finsler-Geometric Flow

Abstract

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This paper considers a compact Finsler manifold $(M^n, F(t), m)$ \nevolving under a Finsler-geometric flow and establishes global gradient \nestimates for positive solutions of the following nonlinear heat \nequation
$$\\partial_{t}u(x,t)=\\Delta_{m} u(x,t),~~~~~~~~~~(x,t)\\in M\\times[0,T],$$<\/p>

where\n $\\Delta_{m}$ is the Finsler-Laplacian. By integrating the gradient \nestimates, we derive the corresponding Harnack inequalities. Our results\n generalize and correct the work of S. Lakzian, who established similar \nresults for the Finsler-Ricci flow. Our results are also natural \nextension of similar results on Riemannian-geometric flow, previously \nstudied by J. Sun.\u00a0 Finally, we give an application to the \nFinsler-Yamabe flow.<\/p>"