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

\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) .\ $Givenfourarbitraryfinitenumbers$r<\alpha <\beta<s,$onecanconstructaradial$\varphi\in C^{0}\left( $\mathbb{R}$% ^{n}\right) \bigcap L^{\infty}\left( $\mathbb{R}$^{n}\right) $sothat$\varphi\ $togetherwithitscorrespondingsolution$\ 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}