On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow

We study the motion of noncompact hypersurfaces moved by their mean curvature obtained by a rotation around x-axis of the graph a function y = u(x, t) (defined for all x ∈ R). We are interested to estimate its profile when the hypersurface closes open ends at the quenching (pinching) time T. We estimate its profile at the quenching time from above and below. We in particular prove that u(x, T)∼ |x|^→aas |x| → ∞ if u(x, 0) tends to its infimum with algebraic rate |x|^-2a (as |x| → ∞ with a > 0).