  Motivated by experimental observations of patterning at the leading edge of
motile eukaryotic cells, we introduce a general model for the dynamics of
nearly-flat fluid membranes driven from within by an ensemble of activators. We
include, in particular, a kinematic coupling between activator density and
membrane slope which generically arises whenever the membrane has a
non-vanishing normal speed. We unveil the phase diagram of the model by means
of a perturbative field-theoretical renormalization group analysis. Due to the
aforementioned kinematic coupling the natural dynamical scaling is acoustic,
that is the dynamical critical exponent is 1. However, as soon as the the
normal velocity of the membrane is tuned to zero, the system crosses over to
diffusive dynamic scaling in mean field. Distinct critical points can be
reached depending on how the limit of vanishing velocity is realised: in each
of them corrections to scaling due to nonlinear coupling terms must be taken
into accounts. The detailed analysis of these critical points reveals novel
scaling regimes wich can be accessed with perturbative methods, together with
signs of strong coupling behaviour, which establishes a promising ground for
further non-perturbative calculations. Our results unify several previous
studies on the dynamics of active membrane, while also identifying nontrivial
scaling regimes which cannot be captured by passive theories of fluctuating
interfaces and are relevant for the physics of living membranes.
