Abstract
With the help of heat equation, we first construct an example of a graphical solution to the curve shortening flow. This solution y (x, t) has the interesting property that it converges to a log-periodic function of the form A sin (log t) + B cos (log t) as t → ∞, where A, B are constants. Moreover, for any two numbers α < β, we are also able to construct a solution satisfying the oscillation limits lim inf y (x, t) = α, lim sup y (x, t) = β, x ∈ K t→∞ t→∞ on any compact subset K ⊂ R.