On an asymptotically log-periodic solution to the graphical curve shortening flow equation

With the help of heat equation, we first construct an example of a graphical solution to the curve shortening flow. This solution $ y\left(x, t\right) \ $has the interesting property that it converges to a log-periodic function of the form $ A\sin \left( \log t\right) +B\cos \left( \log t\right) $ as$ \ t\rightarrow \infty, \ $where $ A, \ B $ are constants. Moreover, for any two numbers $ \alpha < \beta, \ $we are also able to construct a solution satisfying the oscillation limits $ \liminf\limits_{t\rightarrow \infty}y\left( x,t\right) = \alpha,\ \ \ \limsup\limits _{t\rightarrow \infty}y\left( x,t\right) = \beta,\ \ \ x\in K $ on any compact subset$ \ K\subset \mathbb{R}. $