Hamilton-type gradient estimates for Yamabe-type equations on Finsler manifolds

In this paper, we study the positive solution to the Finslerian Yamabe-type equation

$$u_t=\Delta ^{\nabla u} u+au+bu^\alpha .$$

We give the Hamilton-type gradient estimate on compact Finsler metric measure spaces with the celebrated \(CD(-K,N)\) condition. Besides, on forward complete noncompact Finsler metric measure spaces with the mixed weighted Ricci curvature bounded below, the new comparison theorem established by the third author (Shen in Operators on nonlinear metric measure spaces I: A new Laplacian comparison theorem on Finsler manifolds and a traditional approach to gradient estimates of Finslerian Schrödinger equation arXiv:2312.06617v2 [math.DG], 2024) allows us to give the gradient estimate under the assumption of certain bounded non-Riemannian tensors. Finally, we prove the Liouville-type theorem and the Harnack inequality for such solutions as applications.