Type II singularities in area-preserving curvature flows of convex symmetric immersed closed plane curves

We deal with the area-preserving curvature flow in the plane, particularly the blow-up phenomena of curvatures on cusp singularities in contractions of convex immersed curves with self-crossing points. For Abresch-Langer type curves with highly symmetric properties, it has been known that the maximum of curvatures blows up at a finite time under some assumptions. In this paper, we consider the blow-up rates in this case.