Critical behavior of an ignition model in chemical combustion

A model for the hot slab ignition problem is analyzed to determine critical conditions. The system is said to be super-critical if the solution of the reduced perturbation problem blows up in small finite time or sub-critical if the blow up time is large. Comparison principles for integral equations are used to construct upper and lower solutions of the equation. All solutions depend on two parameters ${\varepsilon ^{ - 1}}$, the Zeĺdovitch number and $\lambda$, the scaled hot slab size. Upper and lower bounds on a ’critical’ curve ${\lambda _c}\left ( \epsilon \right )$ in the $\left ( {\epsilon , \lambda } \right )$ plane, separating the super-critical from the sub-critical region, are derived based upon the lower and upper solution behavior. Numerical results confirm the parameter space analysis.