Non-convergence and instability in the asymptotic behaviour of curves evolving by curvature
Let \(\gamma_0\) be a convex closed curve in the plane \(C\). Its motion is described by the equation of the form \[ \frac{\partial}{\partial t} x(\xi,t)=-\phi({\mathbf n}(\xi,t))(\kappa(\xi,t))^{\alpha} {\mathbf n}(\xi,t),\quad x(\xi,0)=x_0(\xi), \] for all \(\xi\) in \(C\) and \(t\in[0,T)\), where \(x_t=x(\cdot,t)\) is a smooth embedding for each \(t\in[0,T), {\mathbf n}(\xi,t)\) is the outward unit normal vector to the curve \(\gamma_t=x_t(C)\) at the point \(x(\xi,t),\kappa(\xi,t)\) is the curvature of \(\gamma_t\) at \(x(\xi,t)\) and \(\phi:S^1\to R\) is a smooth positive function. The author proves that the isoparametric ratios of the evolving curves generically approach infinity if the speed of motion is proportional to curvature to a power less than \(1/3\).
differential equations asymptotics behaviour
scientific article; zbMATH DE number 1893852