Mathematical Analysis of Thermal Runaway for Spatially Inhomogeneous Reactions

Abstract

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 $\delta $, or equivalent, is greater than the upper bound $\delta ^ * $ 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( {\delta - \delta ^ * } )^{ - 1 /2} $ as $\delta $ approaches $\delta ^ * $ from above. For $\delta < \delta ^ * $ lower bounds on the initial data are determined above which thermal runaway must also occur.