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

Abstract

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