The behavior of stationary and moving spatially-periodic patterns in a simple cross-flow reactor was simulated and analyzed for a situation in which reactant is supplied continuously along the reactor and a first-order exothermic reactor occurs. The Danckwerts boundary conditions for realistic Le and Pe values. While the unbounded (infinitely long) reactor is an asymptotic case used to study the stability of the homogeneous solution, the moving waves that emerge in the convectively unstable unbounded system may be arrested at the boundaries of a bounded system and stationary waves are established above some amplification threshold. Sustained periodic and aperiodic behavior may emerge under certain conditions. The spatial behavior in the bounded system with Pe → ∞ is analogous to the temporal behavior of the simple thermokinetic CSTR problem and the behavior of the distributed system is classified according to that of the lumped one.