A prototype chemical reaction is examined in both one and two-dimensional continuous-flow microwave reactors, which are unstirred so the effects of diffusion are important. The reaction rate obeys the Arrhenius law and the thermal absorptivity of the reactor contents is assumed to be both temperature- and concentration-dependent. The governing equations consist of coupled reaction-diffusion equations for the temperature and reactant concentration, plus a Helmholtz equation describing the electric-field amplitude in the reactor. The Galerkin method is used to develop a semi-analytical microwave reactor model, which consists of ordinary differential equations. A stability analysis is performed on the semi-analytical model. This allows the stability of the system to be determined for particular parameter choices and also allows any regions of parameter space in which Hopf bifurcations (and hence periodic solutions called limit-cycles) occur to be obtained. An excellent comparison is obtained between the semi-analytical and numerical solutions, both for the steady-state solution and for time-varying solutions, such as the limit-cycle.