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).
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