  This paper concerns the existence of transonic shocks for steady
exothermically reacting Euler flows in an almost flat nozzle with the small
rate of the exothermic reaction. One of the key points is to quantitatively
determine the position of the shock front in the nozzle. We focus on the
contributions of the perturbation of the flat nozzle and the exothermic
reaction in determining the position of the shock front by comparing the orders
of $\sigma$ and $\kappa$, where $\sigma$ represents the scale of the
perturbation of the flat nozzle and $\kappa$ the rate of the exothermic
reaction. To this end, a free boundary problem for the linearized reacting
Euler system is proposed to catch an approximating position of the shock front
as well as the associated approximating shock solution, with which the
existence of a shock solution close to it can be established via a nonlinear
iteration scheme. One of the key steps is solving the nonlinear equation
derived from the solvability condition for the elliptic sub-problem in the
domain of the subsonic flow behind the shock front, which determines the free
boundary of the proposed problem. Four typical cases are analyzed which
describe possible interactions between the geometry of the nozzle boundary and
the exothermic reaction. The results also manifest that exothermic reaction has
a stabilization effect on transonic shocks in the nozzles.
