  In this paper, we explore the positive solutions to the Finslerian nonlinear
equation
  $$\frac{\partial u}{\partial t} = \Delta^{\nabla u} u + au\log u + bu,$$
  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, using a new
comparison theorem developed by the first author, we also establish a local
gradient estimate on a non-compact forward complete Finsler metric measure
spaces with the mixed weighted Ricci curvature bounded below, as well as finite
bounds of misalignment and some non-Riemannian curvatures. Lastly, we prove the
Harnack inequalities and a Liouville-type theorem of such solutions.
