Based on a general theory of detonation waves with an embedded sonic locus that we have previously developed, we carry out asymptotic analysis of weakly curved slowly varying detonation waves and show that the theory predicts the phenomenon of detonation ignition and failure. The analysis is not restricted to near Chapman–Jouguet detonation speeds and is capable of predicting quasi-steady, normal detonation shock speed versus curvature ($D$–$\kappa$) curves with multiple turning points. An evolution equation that retains the shock acceleration, $\skew2\dot{D}$, namely a $\skew2\dot{D}$–$D$–$\kappa$ relation is rationally derived which describes the dynamics of pre-existing detonation waves. The solutions of the equation for spherical detonation are shown to reproduce the ignition/failure phenomenon observed in both numerical simulations of blast wave initiation and in experiments. A single-step chemical reaction described by one progress variable is employed, but the kinetics is sufficiently general and is not restricted to Arrhenius form, although most specific calculations are performed for Arrhenius kinetics. As an example, we calculate critical energies of direct initiation for hydrogen–oxygen mixtures and find close agreement with available experimental data.