  \begin{abstract}
  Motivated by a classical stabilization result for solution to the Cauchy
  problem of the heat equation$\ \partial_{t}u=\bigtriangleup u\ $on
  $\mathbb{R}^{n}$, we consider its oscillation behavior with radial initial
  data $\varphi\left( x\right) =\varphi\left( \left\vert x\right\vert
  \right) \in C^{0}\left( \mathbb{R}^{n}\right) \bigcap L^{\infty}\left(
  \mathbb{R}^{n}\right) .\ $Given four arbitrary finite numbers $r<\alpha
  <\beta<s,$ one can construct a radial $\varphi\in C^{0}\left( \mathbb{R}%
  ^{n}\right) \bigcap L^{\infty}\left( \mathbb{R}^{n}\right) $ so that
  $\varphi\ $together with its corresponding solution$\ u\left( x,t\right) $
  satisfy the oscillation behavior:
  \begin{align*}
  \liminf_{\tau\rightarrow\infty}\varphi\left( \tau\right) & =r<\liminf
  _{t\rightarrow\infty}u\left( 0,t\right) =\alpha
  & <\limsup_{t\rightarrow\infty}u\left( 0,t\right) =\beta<\limsup
  _{\tau\rightarrow\infty}\varphi\left( \tau\right) =s.
  \end{align*}
  Another related topic concerning the oscillation behavior of the average
integral of the initial data is also discussed.
  \end{abstract}
