Asymptotic behaviors of star-shaped curves expanding by \(V=1-K\). (English) Zbl 1212.53094

Summary: We consider asymptotic behaviors of star-shaped curves expanding by \(V=1-K\), where \(V\) denotes the outward-normal velocity and \(K\) curvature. In this paper, we show the followings. The difference of the radial functions between an expanding curve and circle has its asymptotic shape as \(t\to +\infty \). For two curves, if the asymptotic shapes are identical, then the curves are also. The set of all asymptotic shapes is dense in \(C(S^1)\).


53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35K93 Quasilinear parabolic equations with mean curvature operator