Contracting convex immersed closed plane curves with slow speed of curvature

The authors study the contraction of a convex immersed plane curve with speed 1 α k α \frac {1}{\alpha }k^{\alpha } , where α ∈ ( 0 , 1 ] \alpha \in (0,1] is a constant, and show that, if the blow-up rate of the curvature is of type one, it will converge to a homothetic self-similar solution . They also discuss a special symmetric case of type two blow-up and show that it converges to a translational self-similar solution . In the case of curve shortening flow (i.e., when α = 1 \alpha =1 ), this translational self-similar solution is the familiar “ Grim Reaper ” (a terminology due to M. Grayson).