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/αk ^α, where α ∈ (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), this translational self-similar solution is the familiar "Grim Reaper" (a terminology due to M. Grayson).