text: For , the system explodes since the exponential term dominates as time proceeds. We do not need to wait for a long time for  to blow up. Because of the exponential forcing,  at a finite value of . This time is interpreted as the ignition time or induction time of the system. When , the heat conduction term  can be neglected in which case the problem admits an explicit solution,
  At time , the system explodes. This time is also referred to as the adiabatic induction period since the heat conduction term  is neglected.
  In the near-critical condition, i.e., when , the system takes very long time to explode. The analysis for this limit was first carried out by Frank-Kamenetskii., although proper asymptotics were carried out only later by D. R. Kassoy and Amable Liñán including reactant consumption because reactant consumption is not negligible when .  A simplified analysis without reactant consumption is presented here. Let us define a small parameter  such that . For this case, the time evolution of  is as follows: first it increases to steady-state temperature value corresponding to , which is given by  at times of order , then it stays very close to this steady-state value for a long time before eventually exploding at a long time. The quantity of interest is the long-time estimate for the explosion. To find out the estimate, introduce the transformations  and  that is appropriate for the region where  stays close to  into the governing equation and collect only the leading-order terms to find out
  where the boundary condition is derived by matching with the initial region wherein . The solution to the above-mentioned problem is given by
  which immediately reveals that  when  Writing this condition in terms of , the explosion time in the near-critical condition is found to be
  which implies that the ignition time  as  with a square-root singularity.