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We consider a class of degenerate parabolic equations with linear growth Lagrangian. Two prototypes within this class, sharing common features with nonlinear transport equations, are the relativistic porous medium equation and the speed-limited (or flux-limited) porous medium equation. In arbitrary space dimension, we prove that entropy solutions to the Cauchy problem satisfy the finite speed of propagation property. For the two aforementioned prototypes, we provide a condition on the growth of the initial datum which guarantees the occurrence of a waiting-time phenomenon; we also present a heuristic argument in favor of the optimality of such condition

Finite Speed of Propagation and Waiting Time Phenomena for Degenerate Parabolic Equations with Linear Growth Lagrangian