We show that a certain non-local curvature flow for planar curves has non-trivial self-similar solutions with -fold rotational symmetry, bifurcated from a trivial circular solution. Moreover, we show that the trivial solution is stable with respect to perturbations which keep the geometric center and the enclosed area, and that, for different from 3, the -fold symmetric solution is stable with respect to perturbations which satisfy the same conditions as above and have the same symmetry as the solutions.
Cite this article
Takeo Sugai, Bifurcations of spherically asymmetric solutions to an evolution equation for curves. Interfaces Free Bound. 24 (2022), no. 2, pp. 287–306DOI 10.4171/IFB/474