  In this paper we prove some Hamilton type and Li-Yau type gradient estimates
on positive solutions to generalized nonlinear parabolic equations on smooth
metric measure space with compact boundary. The geometry of the space in terms
of lower bounds on the weighted Bakry-Emery Ricci curvature tensor and weighted
mean curvature of the boundary are key in proving generalized local and global
gradient estimates. Various applications of these gradient estimates in terms
of parabolic Harnack inequalities and Liouville type results are discussed.
Further consequences to some specific models informed by the nature of the
nonlinearities are highlighted.
