Abstract We analyze the classical Graetz problem in a tube with an exothermic surface reaction and show that the heat(mass) transfer coefficient is not a continuous function of the axial position and jumps from one asymptote to another at ignition/extinction points. We show that the steady-state heat(mass) transfer coefficient is not a unique function of position in parameter regions in which the Graetz problem with surface reaction has multiple solutions. We also analyze the more general two-dimensional model (with axial conduction/diffusion included and Danckwerts boundary conditions) and show that for fixed values of the reaction parameters, the heat(mass) transfer coefficient has three asymptotes. Unlike the Graetz problem, in this case the heat(mass) transfer coefficient is always finite and bounded at the inlet and is given by a new asymptote. We present analytical expressions for all three asymptotes for the case of flat and parabolic velocity profiles. It is also shown that in catalytic monoliths, ignition/extinction may often occur in the entry region and hence the local transfer coefficients and not the average values proposed in the literature determine the ignition/extinction behavior. Finally, we use the new results to develop and analyze an accurate one-dimensional two-phase model of a catalytic monolith with position dependent heat and mass transfer coefficients and determine analytically the dependence of the ignition/extinction locus on various design and operating parameters.