Global Existence for Reaction‐Diffusion Systems Modelling Ignition

The pair of parabolic equations , , with a>0 and b>0 models the temperature and concentration for an exothermic chemical reaction for which just one species controls the reaction rate f. Of particular interest is the case where , which appears in the Frank‐Kamenetskii approximation to Arrhenius‐type reactions. We show here that for a large choice of the nonlinearity f(u,v) in (1), (2)(including the model case (3)), the corresponding initial‐value problem for (1), (2) in the whole space with bounded initial data has a solution which exists for all times. Finite‐time blow‐up might occur, though, for other choices of function f(u,v), and we discuss here a linear example which strongly hints at such behaviour. (Accepted September 16, 1996)