The phenomenological equation of spin combustion (singularly perturbed nonlinear parabolic equation of Van-der-Pol’s type with Neumann conditions) is considered in the annular regions (circle, ring, thin ring, and circumference). The action of pseudo-differential operator \((-\Delta )^{\alpha }\) , \(0<\alpha <1\) is determined by spectral decomposition. Spectral problems for the given equation in the circle and in the ring, corresponding quasi-normal forms and attractors, are investigated. Auto-oscillating modes of the problem in the circle and its periodic solutions of the rotating wave type are presented. The character of stability is clarified by the analysis of two- and four-mode approximations. For comparison, using the Krylov-Bogolyubov-Mitropolsky-Samoilenko method, an asymptotic of the created solution of the rotating wave type, when the zone of combustion reaction spread is the ring, is constructed, and the structure and stability of the solutions are investigated. The theorem about the properties of the stability of stationary solutions of quasi-normal form of the problem and the stability of rotating waves of the initial-boundary value problem with Neumann conditions has been proved. The interconnection of the model of spin combustion in the ring region with the model of spin combustion in the circumference is obtained.
