| Summary: | In this article, we study the evolution of immersed locally convex plane curves driven by anisotropic flow with inner normal velocity V=1αψ(x)καV=\frac{1}{\alpha }\psi \left(x){\kappa }^{\alpha } for α<0\alpha \lt 0 or α>1\alpha \gt 1, where x∈[0,2mπ]x\in \left[0,2m\pi ] is the tangential angle at the point on evolving curves. For −1≤α<0-1\le \alpha \lt 0, we show the flow exists globally and the rescaled flow has a full-time convergence. For α<−1\alpha \lt -1 or α>1\alpha \gt 1, we show only type I singularity arises in the flow, and the rescaled flow has subsequential convergence, i.e. for any time sequence, there is a time subsequence along which the rescaled curvature of evolving curves converges to a limit function; furthermore, if the anisotropic function ψ\psi and the initial curve both have some symmetric structure, the subsequential convergence could be refined to be full-time convergence.
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