Classification of steady solutions of the full kinematic model

The geometric type of kinematic model governs any moving plane curve which propagates in the normal direction in the medium. There have appeared some results in the literature via this model. However, these results are all based upon some simplification on the model or prior phenomenological assumptions on the solutions. In this paper, we use really full kinematic model to classify steady solutions (i.e., curvature is independent of time) with positive (or negative) curvature. Various type of solutions have been obtained, such as rotating and translating waves, especially for the rotating spiral waves. Spiral waves arise in many biological, chemical, and physiological systems. The kinematical model can be used to describe the motion of the spiral arms approximated as curves in the plane. In fact, using our results (Theorem 3.8), we can answer the following questions: Is there any steadily rotating wave for a given weakly excitable medium? If yes, what kind of information we can know about these rotating waves? e.g., the tip’s curvature, the tip’s tangential velocity, and the rotating frequency. Comparing our results with previous ones in the literature, there are some differences between them. The only evolving plane curves with positive (or negative) curvature must have monotonous curvatures via simplified model but full model admits solutions with any given oscillating number of the curvatures.