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The evolution of immersed locally convex plane curves driven by anisotropic curvature flow. (English) Zbl 1497.35055

Summary: In this article, we study the evolution of immersed locally convex plane curves driven by anisotropic flow with inner normal velocity \(V=\frac{1}{\alpha} \psi (x) \kappa^{\alpha}\) for \(\alpha < 0\) or \(\alpha > 1\), where \(x\in [0,2m\pi]\) is the tangential angle at the point on evolving curves. For \(-1\leq \alpha < 0\), we show the flow exists globally and the rescaled flow has a full-time convergence. For \(\alpha < -1\) or \(\alpha > 1\), 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 \(\psi\) and the initial curve both have some symmetric structure, the subsequential convergence could be refined to be full-time convergence.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
35K55 Nonlinear parabolic equations
35K93 Quasilinear parabolic equations with mean curvature operator
53E10 Flows related to mean curvature
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