On asymptotic forms of reactive-diffusive runaway

Abstract Different aspects of reactive-diffusive runaway are discussed in this paper. It is shown that a wide range of reactive nonlinearities can be analysed in essentially the same way, leading to the conclusion that there appear to be only two distinct non-singular, self-similar or approximately self-similar forms of description for non-homogeneous blowup. These exactly self-similar and asymptotically self-similar descriptions have previously been recognized, but are examined here in some detail for many different types of nonlinearity using asymptotic and numerical techniques. It is confirmed that exactly self-similar descriptions only behave non-pathologically in fairly extreme, discrete cases that are very close to a reactive-diffusive (or steady-state) balance under three or more dimensions of symmetry. Quantitative identification of these cases indicates that they may be of dubious practical relevance. It is then shown that only one class of non-singular, asymptotically self-similar descriptions can be found using coordinate perturbation techniques. Resulting solutions are derived to a high asymptotic order and are found to reveal a fairly universal structure for symmetric blowup. A simple analysis reveals that non-symmetric blowup also fits in with this structure, at least to leading order.