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\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 averageintegral of the initial data is also discussed.  \end{abstract}

Prescribing Oscillation Behavior of Solutions to the Heat Equation on $\mathbb{R}^n$ via the Initial Data and its Average Integral