  A minimal model for curved detonations is studied, illustrating the role of
the reaction rate on the detonation speed and its propagation limits. The model
is based on a simple extension of the minimal Fickett toy model for detonations
based on the kinematic wave equation. The use of a simple depletion rate
conditioned on the shock speed serves to illustrate its role in the
quasi-steady structure of curved waves and their initiation from a strong blast
wave. Calculations of strong initiation from a self-similar explosion
illustrate the various asymptotic regimes of the transition to self-sustenance
and their link to the steady wave structure. We recover the asymptotic regimes
of detonation formation suggested by He and Clavin and modelled in the context
of Detonation Shock Dynamics by Stewart and collaborators. Following an
analysis using the shock change equation, we identify a unique criterion that
permits to infer the critical energy for initiation from the competition
between energy release and geometric decay.
