In the theoretical study of combustion, ignition has come to mean a wide variety of things, leading to a surprisingly multifaceted range of analyses. The earliest approaches, of Semenov and Frank-Kamenetskii, considered only time or spatial dependendence, one revealing dynamical aspects and the other revealing critical conditions for self-ignition or auto-ignition. These approaches are still alive and being generalised; the identification of inertial manifolds in self-igniting systems can provide some interesting details in the range of possible dynamics. At the heart of the spatially dependent dynamics of self-ignition is a blowup problem in semilinear PDEs, the analysis of which leads on to a complete matching sequence of asymptotic problems describing the transition from slow and benign chemistry to self-propagating flames (some flames are propagated, rather than self-propagating). Ignition in the presence of a point source of heat, a form of forced ignition, encounters another type of critical boundary, the Zeldovich flame ball. The presentation will review this variety of ways in which ignition can be interpreted and studied mathematically.
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