On a nonlinear parabolic equation arising from anisotropic plane curve evolution

We study the asymptotic behavior of a nonlinear parabolic equation arising from anisotropic plane curve evolution. We show that, if the solution has type-one blow-up, then it will converge (after rescaling) to a self-similar solution. If the solution has type-two blow-up, then its profile near the maximum point is also known. In the case of anisotropic plane curve evolution, if the evolving curve has type-two blow-up, then it will converge to a translational self-similar solution.