In this manuscript, we explore the positive solutions to the Finslerian nonlinear equation \(\begin{aligned} \frac{\partial u}{\partial t} = \Delta ^{\nabla u} u + au\log u + bu, \end{aligned}\) which is related to Ricci solitons and serves as the Euler-Lagrange equation to the Finslerian log-energy functional. We then obtain the global gradient estimate of its positive solution on a compact Finsler metric measure space with the weighted Ricci curvature bounded below. Furthermore, applying a more general method based on a new comparison theorem established by the first author, we derive the local gradient estimate on non-compact forward complete Finsler metric measure spaces with the mixed weighted Ricci curvature bounded below, under additional assumption of finite bounds of misalignment and some non-Riemannian curvatures. Lastly, we prove the Harnack inequalities for such solutions, as well as boundness or gap property of global solutions.
