We study the contraction of a convex immersed plane curve with speed 1 α k 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. We 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 [GR]). * AMS Subject Classifications: 35K15, 35K55. We thank Ben Andrews for giving us this summary.rescaled about the final point (the isoperimetric ratio approaches infinity); and the exceptional ones where the isoperimetric ratio remains bounded converge to homothetic solutions, which have been classified. For α > 1/3, the rescaled solutions converge to circles; and for α = 1/3, they converge to ellipses.Remark 2 As a consequence of Theorem 1, we have the following interesting elliptic result. For 0 < λ < 3 (here λ = 1/α), the only positive 2π-periodic solution to the equation
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