Mathematical Analysis of Thermal Runaway for Spatially Inhomogeneous Reactions

Parabolic initial value problems, including the Frank–Kamenetskii approximation for a high-activation energy exothermic chemical reaction, are studied. It is found that for many such systems there is finite time blow-up of the solution whenever the Frank–Kamenetskii parameter δ, or equivalent, is greater than the upper bound δ ^ * to the spectrum of the corresponding steady state. When the upper bound lies in the spectrum the blow-up time is found to increase as O( δ - δ ^ * )- 1 /2 as δ approaches δ ^ * from above. For δ < δ ^ * lower bounds on the initial data are determined above which thermal runaway must also occur.